- 


REESE    LIBRARY 

-1       Itll: 

NIVERSiTY   OF   CALIFORNIA. 


rS8 


Accessions  No.  ^  ^  ^2  3     Sheh  A ' 


KINEMATICS. 


A  TREATISE  ON  THE  MODIFICATION  OF  MOTION,  AS  AFFECTED  BY 

THE  FORMS  AND  MODES  OF  CONNECTION  OF  THE 

MOVING    PARTS    OF  MACHINES. 


FULLY     ILLUSTRATED 

BY  ACCURATE  DIAGRAMS  OF  MECHANICAL  MOVEMENTS, 
AS   PRACTICALLY  CONSTRUCTED  ; 


FOR  THE   USE    OF   DRAUGHTSMEN,    MACHINISTS,    AND    STUDENTS   OF 
MECHANICAL  ENGINEERING. 


BY 

CHARLES  WILLIAM  MAC  CORD,   A.M.,  Sc.D., 

PROFESSOR  OF  MECHANICAL  DRAWING  IN  THE  STEVKN8  INSTITUTE  OF  TECHNOLOGY,  HOBOKEN,  II.  J. 

AUTHOR  OF  "LESSONS  IN    MECHANICAL    DRAWING,"     "A  PRACTICAL    TREATISE    ON    THE 

BLIDE  VALVE  AND  ECCENTRIC,"  AND  VARIOUS  MONOGRAPHS  ON  MECHANISM. 


UNIVERSITY 


NEW  YORK : 

JOHN    WILEY    &    SONS. 


1883. 


COPYRIGHT,  1883, 
BY  CHAS.  WILLIAM  MAC  CORD. 


Snscribeb 

TO  THE  REV.  JOHN  T.   DUFFIELD,  D.D.,  LL.D., 

OF  PRINCETON^  COLLEGE, 
BY   HIS  FRIEND  AND  FORMER  PUPIL, 


PREFACE. 


A  word  of  explanation  is  due  to  the  reader,  in  view  of  the  fact  that 
the  following  pages  relate  to  but  a  small  number  of  the  vast  array  of 
devices  included  in  the  broad  term  Mechanism. 

Having  in  the  opening  chapters  considered  the  methods  by  which 
motion,  and  the  modification  of  motion,  may  be  represented  and  an- 
alyzed, and  the  basis  upon  which  a  proper  classification  of  elementary 
mechanical  combinations  may  be  made,  the  next  question  was,  what 
classes  of  such  combinations  should  be  first  examined.  So  large  a 
proportion  of  these  consist  of  pieces  rotating  in  contact  about  fixed 
axes,  that  they  seemed  to  have  a  natural  claim  to  precedence,  which 
was  duly  recognized. 

Attention  was  accordingly  next  directed  to  the  discussion  of  the 
pitch  surfaces,  and  in  natural  sequence  to  the  forms  of  the  teeth,  of 
gear  wheels  of  all  kinds.  Which  having  been  done,  it  appeared 
proper  to  publish  so  much  as  was  completed,  because  notwithstanding 
that  many  treatises  upon  these  special  topics  have  been  written,  there 
would  seem  to  be  room  for  another ;  the  more  particularly  since  even 
in  sweeping  out  this  part  of  the  shop,  scraps  of  new  material  and  cut- 
tings of  old  have  been  found,  in  quantity  and  of  quality  to  be  worth 
using. 

The  endeavor  has  been  made  to  treat  the  theory  of  the  subject  in  a 
practical  manner,  for  the  benefit  of  the  practical  man.  That  is  to 
say,  the  demonstrations  are  made  as  far  as  possible  directly  dependent 
upon  the  diagrams ;  and  the  latter,  in  most  cases  reduced  from  work 
actually  executed  upon  a  large  scale,  are  accompanied  by  explanations 
which  it  is  hoped  will  enable  any  ordinarily  expert  draughtsman  to 
"lay  out"  the  movements  with  ease  and  accuracy. 

In  order  to  avoid  interrupting  the  argument  by  subordinate  discus- 
sions, as  well  as  for  more  ready  reference,  an  Appendix  has  been 
added,  containing  the  methods  of  construction,  and  other  graphic 


Till  CONTEXTS. 

CHAPTER  VI. 

TAGE 

Tooth  Gearing— 1.  Classification  of  Toothed  Gearing.— 2.  Spur  Wheels.— 
Epicycloidal  Teeth,  Outside  Gear,  Generation  of  the  Outline. — Pitch, 
Angles  of  Action,  Backlash,  Clearance,  Approaching  and  Receding 
Action. — Interchangeable  Wheels. — Size  of  Describing  Circle. — Rack  and 
Pinion. — 3.  Inside  Spur  Gearing.  —Intermediate  Describing  Circle. — Limit- 
ing Diameters  of  Generating  Circles 88 

CHAPTER  VII. 

Low-Numbered  Pinions — Practical  Limit  of  Obliquity — Pinions  with  3,  4,  and 
5  Leaves — Least  Number  that  can  be  Used  with  a  Given  Driver — The  Two- 
Leaved  Pinion — Low  Numbers  in  Inside  Gear — Various  Examples — Infe- 
rior and  Superior  Limits — Two-Leaved  Pinions  in  Inside  Gear — Least  Num- 
ber that  can  be  Used  with  a  Given  Follower 110 

CHAPTER  VIII. 

Spur  Gearing,  Continued — Epicycloidal  System. — Limiting  Numbers  of  Teeth 
for  Various  Arcs  of  Action.  Details  of  Trigonometrical  Process  of  Deter- 
mination. The  Nomodont  or  Curve  of  Limiting  Values 128 

CHAPTER  IX. 

Spur  Gearing,  Continued — Involute  Teeth. — Involute  Generated  by  Rolling  of 
Right  Line  on  Base  Circles.  Peculiar  Properties.  Original  Pitch  Circle. 
Rack  and  Pinion.  Annular  Wheels.  Low-numbered  Pinions.  Involute 
Tooth  with  Epicycloidal  Extensions.  Limiting  Numbers  for  Given  Arcs 
of  Recess.  Comparison  of  the  Involute  and  Epicycloidal  Systems.  Invo- 
lute Generated  by  Rolling  of  Logarithmic  Spiral  on  Pitch  Circles 143 

CHAPTER  X. 

Spur  Gearing,  Continued. — Conjugate  Teeth.  Sang's  Theory.  Path  of  Con- 
tact. Unsymmetrical  Teeth.  Approximate  Forms.  The  Odontographs. 
Diametral  Pitch.  Manufacture  of  Gear  Cutters.  Determination  of  Series 
of  Equidistant  Cutters 167 

CHAPTER  XI. 

1.  Twisted  Spur  Gearing.— Hooke's  Stepped  Wheels.  Twisted  Wheels.  Rota- 
tion not  due  to  Screw-like  Action.  Elimination  of  Sliding.  Neutraliza- 
tion of  End  Pressure.  2.  Pin  Gearing. — Generation  of  Elementary  Tooth. 
Derivation  of  Working  Tooth.  Peculiarities  of  the  Action.  Racks  and 
Pinions.  Annular  Wheels.  Determination  of  Angle  of  Action.  Deter- 
mination of  Limiting  Numbers  of  Teeth.  Wheels  with  Radial  Planes.  3. 
Non-Circular  Spur  Gearing. — Construction  of  Teeth  for  Elliptical  and 
Lobed  Wheels. .  .  195 


CONTENTS.  ix 

CHAPTER  XII. 

PACK 

1.  The  Teeth  of  Bevel  Wheels.— General  Principles  of  their  Correct  Forma- 
tion by  means  of  a  Describing  Cone.  Tredgold's  Approximate  Method. 
Details  of  Correct  Process.  Results  of  the  two  Methods  Compared,  for 
Epicycloidal  Teeth.  Inside  Bevel  Gearing.  The  Involute  System.  Action 
of  Bevel  and  Spur  Wheels  Compared.  Teeth  of  Conical  Lobed  Wheels. 
Methods  of  Cutting  the  Teeth.  Corliss's  Bevel  Gear  Cutter.  Twisted 

Bevel  Wheels.     2.  The  Teeth  of  Skew  Wheels. Theory  of  Willis  and 

Rankine.  Describing  Hyperboloid  Approximate  Methods.  Direct  Con- 
struction of  Teeth  thus  Generated.  Fallacy  of  this  Theory.  Such  Teeth 
Impracticable.  A  New  Theory.  Oliviers's  Involute  in  Different  Planes. 
The  Fronts  of  the  Teeth  Single  Curved.  They  Vanish  at  the  Gorge.  The 
Length  of  the  Tooth  Limited.  Backs  of  Teeth  Warped.  The  Conjugate 
Teeth  Unlike.  Action  Reversed  on  Crossing  the  Gorge.  Twisted  Skew 
Wheels.  3.  The  Teeth  of  Screw  Wheels.— Common  Worm  and  Wheel. 
Construction  Referred  to  that  of  Rack  and  Pinion.  Distinctive  Features 
cf  the  Action.  Close-fitting  Tangent  Screw.  Practical  Proportions. 
'  Sang's  Theory  Embodied.  Multiple- threaded  Screw  Wheels.  Screw  and 
Rack.  Oblique  Screw  Gearing.  Construction  of  Teeth  from  Oblique  Rack 
and  Pinion.  Peculiar  Modification  of  the  Action.  Close-fitting  Oblique 
Worm.  Oblique  Screw  and  Rack.  Construction  for  Least  Amount  of 
Sliding.  Resemblance  to  Skew  Gearing.  Hour-glass  Worm  Gear. 
General  Arrangement.  Form  of  the  Pitch  Surface  of  the  Worm.  Forms 
of  the  Threads.  Action  Confined  to  one  Plane.  Forms  of  the  Wheel-teeth. 
Multiple-threaded  Hour-glass  Worm  with  Face-gear  Wheel.  Rollers  Sub- 
stituted for  Teeth  upon  the  Wheel.  4.  The  Teeth  of  Face  Wheels.— Equal 
Wheels  with  Cylindrical  Pins.  Equal  Wheels  with  Axes  perpendicular  to 
each  other.  Unequal  Wheels  Similarly  Situated.  Unequal  Wheels  with 
Axes  in  Different  Planes.  Miscellaneous  Arrangements  of  Face  Wheels. 
Combination  of  Face  and  Screw  Gearing.  Spherical  Screw  and  Wheel. . .  232 

Appendix 295 

Index..,  329 


PRACTICAL  KINEMATICS. 


CHAPTER  I. 


1.  Mechanism  is  the  science  which  treats  of  the  designing  and  con- 
struction of  machinery.     Its  objects  are,  to  investigate  those  abstract 
principles  which  are  involved  in  planning  correctly,  and  to  describe 
the  practical  operations  involved  in  successful  execution. 

2.  A  Machine  is  very  properly  said,  in  a  general  way,  to  be  an  arti- 
ficial work  which  serves  to  apply  or  to  regulate  moving  power. 

This  definition  will  not  answer  our  purpose,  for  which  it  is  not  suf- 
ficiently minute  ;  but  it  is  correct  so  far  as  it  goes,  and  close  and  clear 
enough  for  the  ordinary  employment  of  the  word. 

From  its  terms  we  infer  that  a  machine  applies  or  regulates  ex- 
traneous power  for  some  useful  purpose.  That  is  to  say,  the  existence 
of  the  machine  presupposes  the  existence  of  something  to  be  done 
and  of  power  to  do  it ;  and  it  also  implies  the  necessity  of  modifying, 
in  some  way  or  other,  both  the  force  and  the  motion  caused  by  the 
force.  No  machine  can  move  itself,  nor  can  it  create  motive  power ; 
this  must  be  derived  from  external  sources,  such  as  the  falling  of  a 
weight,  the  uncoiling  of  a  spring,  or  the  expansion  of  steam. 

3.  Motive  power  has   different  characteristics,    according  to  the 
nature  of  the  source.     It  may  be  constant,  as  in  the  case  of  a  head  of 
water  kept  at  the  same  level  by  an  unfailing  stream  ;  it  may  vary  ac- 
cording to  regular  laws,  as  when  derived  from  expanding  steam ;  it 
may  vary  irregularly,  like  the  strength  of  animals  :  or  it  may  be  wholly 
fitful  and  uncertain,  like  the  wind. 

But  these  characteristics,  as  well  as  the  supply  of  power  itself,  are 
beyond  our  control.  We  cannot  create  power  as  we  want  it,  but  must 
take  it  whence  we  can  get  it.  We  cannot  stipulate  conditions,  but 


2  PURE   AND   CONSTRUCTIVE   MECHANISM. 

must  take  the  power  as  we  find  it,  be  thankful  for  it  at  that,  and 
adapt  it  to  our  needs  and  purposes  as  best  we  can. 

4.  This  is  done  by  the  use  of  machinery  :  and  it  is  clear  that  in  the 
construction  of  every  machine,  reference  must  be  had  to  the  charac- 
teristics of  the  motive  power  as  well  as  to  the  nature  of  the  work  to  be 
done.     We  may  therefore  amplify  the  definition  above  given,  and  say 
that— A  Machine  is  an  assemblage  of  moving  parts,  interposed  between 
the  power  and  the  work,  for  the  purpose  of  adapting  the  one  to  the 
other. 

It  is  not,  however,  always  necessary  to  trace  back  the  source  of 
power  to  its  origin.  For  example,  a  line  of  shafting,  whether  itself 
driven  by  wind,  water,  or  steam,  may  properly  be  considered  as  the 
"source  of  power"  in  reference  to  the  various  drills,  lathes,  planers, 
etc.,  driven  by  it. 

Pure  and  Constructive  Mechanism. 

5.  The  operation  of  any  machine  depends  upon  two  things,  viz.  : 
definite  force  and  determinate  motion.     And  in  the  process  of  design- 
ing, due  consideration  must  be  given  to  both  these,  so  that  each  part 
may  be  adapted  to  bear  the  strains  put  upon  it,  as  well  as  to  move 
properly  in  relation  to  other  parts. 

But  the  nature  of  the  movements  does  not  depend  upon  the  strength 
nor  upon  the  absolute  dimensions  of  the  moving  pieces,  and  may  often 
be  clearly  illustrated  by  a  model  whose  proportions  are  very  unlike 
those  of  the  actual  working  machine.  Consequently  the  force  and  the 
motion  may  be  considered  separately  ;  and  thus  the  science  of  Mechan- 
ism is  divided  into  two  branches,  called  respectively  Pure  and  Con- 
structive. 

6.  The  selection  of  materials,  and  the  proportioning  of  details  with 
reference  to  strength  and  durability,  are  governed  directly  by  consid- 
erations relating  to  the  forces  involved.     Closely  connected  with  these 
are  other  considerations  relating  to  facility  in  manufacturing,  con- 
venience in  repairing,  and  kindred  features  essential  to  practical  ex- 
cellence ;  and  the  whole  fall  properly  within  the  scope  of  Constructive 
Mechanism. 

But  we  may  examine  the  action  of  a  machine  by  merely  putting 
it  in  motion,  without  actually  setting  it  at  work ;  and  we  can  plan 
its  movements  without  regard  to  the  requisite  strength  of  the 
parts. 

The  laws  of  motion  may  be  discussed  quite  independently  of  any 
consideration  of  the  force  involved,  and  without  reference  to  either 


MOTION   AND   BEST.  3 

the  power  or  the  work ;  and  this  constitutes  the  branch  called  Pure 
Mechanism. 

7.  Purposing  now  to  confine  our  attention  to  this  latter  branch  of 
the  science,  it  is  necessary,  before  entering  upon  its  study,  to  define 
and  explain  the  sense  in  which  certain  terms  and  phrases  of  frequent 
occurrence  shall  be  used.     And  first  of  all,  we  perceive  that  we  have 
not,  as  yet,  assigned  to  the  word  machine  a  meaning  which  is  precise 
and  in  accordance  with  the  above  limitation. 

We  must  therefore  modify  the  definition  once  more  ;  and  consider- 
ing it  with  reference  to  its  motion  only, 

A  Machine  is  a  combination  of  parts  so  connected  that  when  one 
moves  according  to  a  given  law,  the  others  must  move  according  to  cer- 
tain other  laws. 

8.  Motion  and  Rest  are  essentially  relative  terms,  within  the  limits 
of  our  knowledge.     We  can  conceive  a  body  to  remain  in  a  fixed  posi- 
tion in  space,  but  we  cannot  know  that  there  is  one  which  does.     If 
there  be  any  such  body,  it  is  in  a  state  of  absolute  rest. 

If  two  bodies,  although  both  are  moving  in  space,  retain  the  same 
relative  positions,  each  is  said  to  be  at  rest  with  respect  to  the  other  : 
if  they  do  not,  either  may  be  said  to  be  in  motion  with  respect  to  the 
other. 

Path. — A  point  moving  in  space  describes  a  line,  called  its  path; 
which  may  be  rectilinear  or  curvilinear.  The  motion  of  a  body,  or 
geometrical  magnitude,  may  be  defined  by  the  paths  of  one  or  more  of 
its  points,  selected  at  pleasure. 

Direction. — In  a  given  path,  a  point  can  move  in  either  of  two  direc- 
tions only,  which  may  be  defined  in  various  ways,  as  up  or  down,  to 
the  right  or  left,  with  the  clock  or  the  reverse  ;  direction,  as  well  as 
motion,  being  entirely  relative. 

9.  Velocity,  however,   is  not   essentially  relative.      Whatever  the 
form  of  the  path,  the  speed  of  the  motion  is  estimated  by  comparing 
the  distance,  or  space,  through  which  a  point  or  body  moves,  with  the 
tune  occupied  in  doing  it.     And  since  both  space  and  time  are  abso- 
lute magnitudes,  the  velocity  itself  is  absolute. 

Velocity  is  either  uniform,  equal  spaces  being  traversed  in  equal 
times  ;  accelerated,  the  spaces  increasing,  or  retarded,  the  spaces  de- 
creasing, while  the  times  remain  equal.  And  the  rate  of  acceleration 
or  of  retardation  may  itself  be  either  uniform  or  fluctuating.  But  it 
is  not  necessary  to  consider  all  the  complications  which  may  arise  in 
this  way ;  for  our  purposes  it  will  suffice  to  make  one  general  distinc- 
tion, viz. :  that  between  motions  with  uniform  velocities  and  motions 
with  variable  velocities. 


4  ANGULAK  VELOCITY. 

10.  In  the  case  of  uniform  motion,  the  s^pace  varies  directly  with 
both  the  time  and  the  velocity.  Thus  if  one  body  move  twice  as  long 
and  twice  as  fast  as  another,  it  will  clearly  travel  four  times  as  far. 
This  is  otherwise  expressed  by  saying  that  the  space  increases  in  the 
compound  ratio  of  the  time  and  the  velocity  ;  or,  still  better,  in  the 
form  of  an  equation,  thus, 

Space  =  Time  x  Velocity. 

The  space  and  the  time  are  measured  by  comparing  them  with  fixed 
standards,  or  units,  and  may  therefore  be  expressed  by  abstract  num- 
bers. And  so  in  consequence  may  the  velocity  ;  for  from  the  above 
equation  we  have 


=  3ity' 


in  which  the  first  member  being  composed  of  abstract  terms,  the 
second  member  will  also  be  an  abstract  number,  showing  how  many 
units  of  space  are  traversed  in  a  unit  of  time. 

And  this  is  the  measure  of  absolute  velocity  when  the  motion  is 
uniform. 

11.  Angular  Velocity.  —  This  expression  relates  to  rotatory  motion, 
like  that  of  a  wheel  turning  in  its  bearings  ;  the  speed  of  which  may 
be  measured  by  the  linear  velocity  of  any  point  in  the  rotating  body 
whose  radial  distance  from  the  axis  is  equal  to  the  unit  of  space. 
This  is  called  the  angular  velocity,  and  may  be  either  uniform  or 
variable. 

If  the  angular  velocity  be  uniform,  the  linear  velocity  of  any  point 
varies  directly  as  its  distance  from  the  axis  :  for  the  angles  are  propor- 
tional to  the  times,  and  the  arcs  to  the  radii.  Thus  if  one  point  be 
two  feet,  and  another  four  feet,  from  the  axis,  the  outer  will  move 
twice  as  rapidly  as  the  inner,  since  in  the  same  time  it  describes  twice 
as  large  a  circle. 

The  speed  of  a  wheel  may  also  be  conveniently  expressed  by  stating 
the  number  of  turns  it  makes  in  a  given  time  ;  which  evidently  varies 
as  the  angular  velocity,  if  the  latter  be  uniform. 

The  most  concise  and  useful  value,  however,  is  the  equation 

,     TT  ,     .          Linear  Velocity 
Angular  Velocity  =          Radiug 

12.  Revolution  and  Rotation.—  A  point  is  said  to  revolve  about  a 
right  line  as  an  axis,  when  it  describes  a  circle  of  which  the  centre  is 


VARIOUS   KINDS   OF  MOTION.  5 

in,  and  the  plane  perpendicular  to,  that  line.  ^  When  all  the  points 
of  a  body  thus  revolve,  with  the  same  angular  velocity  and  therefore 
without  changing  their  relative  positions,  the  body  itself  is  said  to  re- 
volve about  the  .axis. 

If  the  axis  passes  through  the  body,  as  in  the  case  of  a  wheel,  the 
word  rotation  may  be  properly  used  synonymously  with  revolution. 

But  it  frequently  occurs  in  mechanical  combinations,  as  for  instance 
in  Watt's  Sun-and-Planet  Wheels,  that  a  body  not  only  rotates  about 
an  axis  which  passes  through  it,  but  at  the  same  time  moves  in  an 
orbit  about  another  axis.  In  order  to  make  a  distinction  between  the 
two  motions,  we  shall  in  such  cases  speak  of  the  first  as  a  Rotation,  and 
of  the  second  as  a  Revolution,  just  as  we  say  that  the  earth  rotates  on 
its  axis,  and  revolves  around  the  sun. 

13.  Continuous  Motion. — Motion  is  in  its  nature  continuous,  in  the 
sense  that  a  point  cannot  move  from  one  position  in  space  to  another, 
without  passing  through  all  the  intermediate  positions,  whether  its 
path  be  rectilinear  or  otherwise.     But  in  the  nature  of  things  it  is  im- 
possible for  a  point  to  go  on  moving  indefinitely  in  the  same  direc- 
tion, unless  its  path  be  one  that  returns  into  itself,  like  a  circle,  ellipse, 
or  other  closed  curve. 

And  the  possibility  of  such  indefinite  continuance  is  what  is  im- 
plied in  the  expression  continuous  motion,  as  technically  employed. 
A  wheel  turning  freely  in  its  bearings  affords  an  example  of  motion 
continuous  in  this  sense,  which  naturally  occurs  oftener  in  circular 
paths  than  in  any  others. 

14.  Reciprocating  Motion. — If  a  point  traverses  the  same  path  alter- 
nately in  opposite  directions,  its  motion  is  called  reciprocating,  whether 
the  path  be  rectilinear  or  not.     But  if  the  point  travel  in  a  circular  or 
other  arc,  the  use  of  this  term  will  be  confined  to  those  cases  in  which 
the  arc  traversed  is  less  than  a  circumference.     For  if  a  wheel  make 
a  number  of  complete  turns,  first  in  one  direction  and  then  in  the 
other,  it  is  manifestly  improper  to  style  such  motion  reciprocating, 
notwithstanding  the  recurring  reversals  in  direction. 

Reciprocating  circular  motion,  like  that  of  a  pendulum,  or  of  a  lever 
swinging  on  a  fixed  centre,  is  also  called  vibration. 

15.  Intermittent  Motion. — When  a  reciprocating  piece  has  reached 
the  end  of  its  excursion  in  one  direction,  there  must  evidently  be  an 
instant  of  rest,  before  it  begins  to  return.     But  it  frequently  is  re- 
quired that  a  piece  shall  remain  still  for  a  definite  time,  after  which 
it  again  moves,  either  in  the  same  direction  as  before  or  in  the  oppo- 
site.    When  a  piece  in  its  action  thus  alternates  motion  with  definite 
periods  of  rest,  it  is  said  to  have  an  intermittent  motion.     If  the  mo- 


6  CYCLE   AND  PHASES   OF  MOTION. 

tions  occur  alternately  in  opposite  directions,  the  action  may  be  called 
.an  intermittent  reciprocating  motion. 

16.  Ifechanical  Movements. — The  different  kinds  of  motion  above 
specified  are  to  a  certain  extent  interchangeable.     That  is  to  say,  one 
kind  may  be  converted  into  another,  by  means  of  various  devices, 
which  are  called  mechanical  movements. 

It  is  ordinarily  the  case,  that  a  machine  is  composed  of  a  number 
of  such  movements,  or  subsidiary  combinations  of  parts,  each  fulfilling 
a  distinct  function  in  the  general  operation.  They  are  to  the  machine 
what  the  members  are  to  the  body ;  but  each  one,  serving  a  definite 
purpose  in  respect  to  its  motion,  may  be  regarded  as  a  little  machine, 
whose  action  may  be  studied  by  itself.  For  example,  the  valve  gear 
of  a  steam  engine  may  be  entirely  disconnected  from  the  other  parts, 
and  its  operation  investigated  without  reference  to  them. 

17.  Cycle  of  Motions. — When  a  mechanical  combination  is  set  in 
action,  its  parts  go  through  a  certain  series  of  motions,  involving 
various  changes  in  direction,  velocity,  or  kind  of  motion,  in  a  regular 
order.     It  is  usually  the  case  that  the  parts  finally  return  to  their 
original  positions,  after  which  the  same  motions  will  recur  in  the  same 
order,  and  so  on  perpetually.     The  whole  series  is  called  the  Cycle. 
Under  these   circumstances  the  combination  is   also  said  to  have  a 
Uniform  Periodic  Motion.     These  terms,  however,  are  used  only  for 
want  of  better  ones.     Neither  the  word  "  Cycle  "  nor  the  word  "  Peri- 
odic "  have  any  reference  to  the  time  required  to  go  through  the  series 
of  motions,  nor  does  the  word  "Uniform"  imply  that  .the  time  occu- 
pied is  always  the  same. 

18.  The  terms  are  intended  to  convey  the  idea  of  regularity  of 
succession  and  constancy  of  relation,  as  obtaining  among  the  motions 
which  make  up  the  scries. 

To  illustrate  :  One  revolution  of  the  crank  of  a  steam  engine  pro- 
duces a  reciprocation  of  the  piston  and  a  series  of  different  angular 
positions  of  the  connecting  rod,  which  itself  vibrates  on  a  moving 
axis.  And  if  the  speed  of  the  crank  be  uniform,  the  velocities  of 
both  piston  and  connecting  rod  will  vary  according  to  a  definite  law. 
Now  it  is  clear  that  the  parts  will  go  through  the  same  series  of 
motions,  in  the  same  relative  order,  and  with  the  same  variations  in 
velocity  as  compared  with  the  speed  of  the  crank,  at  every  turn  of  the 
latter,  whether  it  go  quickly  or  slowly,  uniformly  or  variably,  in  one 
direction  or  the  other. 

19.  Phases  of  Motion. — This  term  is  used  to  designate  the  successive 
phenomena  of  varied  motion.     Thus,  one  phase  in  the  movement  of  a 
steam  engine  would  be  represented  in  a,  diagram  showing  the  relative 


ELEMENTARY   COMBINATIONS.  7 

positions  of  the  piston,  connecting  rod,  and  crank,  at  the  beginning 
of  a  stroke ;  another  if  the  middle  of  the  stroke  had  heen  selected, 
and  so  on.  Such  diagrams,  representing  different  phases  of  the  mo- 
tion, are  often  of  the  greatest  utility  in  conveying  clear  ideas  of  the 
action  of  complicated  mechanical  movements.  Since  the  order  of 
recurrence  is  the  same,  we  may  select  at  pleasure  any  phase  as  the 
beginning  of  a  cycle. 

20.  Recapitulating  a  little,  we  see  from  the  foregoing  that  every 
machine,  regarded  broadly  as  a  contrivance  for  utilizing  power,  con- 
sists essentially  of  three  classes  of  parts ;  the  function  of  the  first 
being  to  receive  the  power,  that  of  the  second  to  transmit  and  modify 
the  force  and  the  motion,  and  that  of  the  third  to  do  the  work.    Evi- 
dently, the  nature  and  form  of  the  first  class  depend  directly  and 
largely  on  the  character  of  the  motive  power,  those  of  the  third  class 
upon  the  nature  of  the  work  to  be  done,  which  also  to  a  great  extent 
determines  the  proper  actual  velocity  of  the  machine  while  in  practical 
operation.     It  is  also  apparent  that  these  three  classes  of  parts  are  in- 
dependent of  each  other,  in  so  far  that  any  kind  of  work  may  be  done 
by  any  kind  of  power,  and  by  means  of  different  combinations  of 
interposed  mechanism. 

21.  Now  in  what  follows,  we  shall  have  to  do  only  with  the  second 
of  these  classes.     Our  object  is  to  investigate  the  laws  which  govern 
the  determinate  motions  involved  in  the  action  of  the  machine  :  and 
the  motions  as  well  as  the  form  of  the  first  class  are  determined  by  the 
manner  of  action  of  the  motive  power,  and  those  of  the  third  by  the 
nature  of  the  work  and  the  manner  in  which  it  is  to  be  done.     The 
motion  of  the  one  class  has  then  to  be  transmitted  to  the  other  ;  and 
as  the  given  motion  of  the  former  may  be  and  usually  is  different  from 
the  required  motion  of  the  latter,  it  follows  that,  during  transmission, 
the  motion  must  be  modified  according  to  specific  conditions.     These 
objects  are  accomplished  by  the  second  of  the  three  classes  of  parts 
above  enumerated  ;   and  it  is  the  province  of  Pure  Mechanism  to  dis- 
cuss the  methods  by  which  motion  may  be  transmitted,  and  to  inves- 
tigate the  laws  which  govern  its  modification  during  the  process. 

22.  Elementary  Combinations. — If  two  pieces  be  so  connected  and 
arranged  that  a  given  motion  imparted  to  one  compels  the  other  to 
move  in  a  determinate  manner,  these  two   constitute  an  elementary 
combination.     Practically,  the  motions  are  made  determinate  by  means 
of  a  rigid  frame,  in  relation  to  which,  as  well  as  in  relation  to  each 
other,  the  two  pieces  move.     But,  obviously,  the  modification  of  the 
motion  is  best  seen  by  comparing  the  movements  of  the  two  pieces  with 
each  other,  so  that  for  our  purposes  it  is  sufficient  to  take  note  of  the 


8  TRANSMISSION   OF   MOTION. 

motion  of  which  each  one  is  capable,  without  regard  to  the  means 
by  which  it  is  restrained  or  limited. 

23.  Driver  and  Follower. — That  piece  of  an   elementary  combina- 
tion to  which  motion  is  supposed  to  be  imparted,  is  called  the  Driver : 
and  the  one  whose  motion  is  made  compulsory  by  the  action  of  the 
other  upon  it,  is  called  the  Follower. 

A  Train  of  Mechanism  consists  of  a  series  of  parts,  composing  the 
whole  or  a  portion  of  a  machine,  each  of  which  receives  motion 
from  the  preceding  one,  and  transmits  it  to  the  next  in  order.  The 
train  is,  therefore,  made  up  of  elementary  combinations,  and  each 
piece  is  a,  follower  to  the  one  which  comes  before,  and  a  driver  to  the 
one  which  comes  after  it.  As  the  motion  may  be  modified  at  each 
successive  step,  it  is  necessary  to  begin  by  considering  the  modifica- 
tions which  may  be  effected  by  means  of  elementary  combinations 
only. 

24.  Modes  of  Transmission. — Strictly  speaking,  if  we  leave  out  of  the 
question  the  agency  of  attractive  or  repulsive  forces,  such  as  magnet- 
ism, one  piece  cannot  compel  another  to  move  unless  the  two  are  in 
actual  contact. 

But  in  many  cases  the  motion  of  one  piece  is  communicated  to 
another  by  the  intervention  of  a  third  one,  under  such  circumstances 
that  the  movements  of  the  latter  are  of  no  possible  consequence,  the 
proper  action  of  the  whole  depending  entirely  on  the  relative  motions 
of  the  first  and  second  :  and  these  two  may  then  be  properly  regarded 
as  forming  an  elementary  combination.  We  have,  then,  that  motion 
may  be  transmitted  from  a  driver  to  a  follower, 

1.  By  Direct  Contact. 

2.  By  Intermediate  Connectors. 

25.  Links  and  Bands. — Such  an  intermediate  connector  must  be  either 
rigid  or  flexible.     If  it  be  rigid,  it  is  called  a  Link,  and  can  either 
push  or  pull,  like  the  connecting  rod  of  a  steam  engine  :  being  neces- 
sarily pivoted  or  otherwise  jointed  to  both  the  driver  and  the  follower. 

If  the  intermediate  connector  be  flexible,  it  is  called  a  Band :  for 
our  purposes  it  is  supposed  to  be  inextensible,  and  it  can  transmit 
motion  only  by  pulling. 

26.  Modification  of  Motion. — In  the  action  of  an  "elementary  combi- 
nation, the  motion  of  the  follower  may  differ  from  that  of  the  driver 
in  kind,  in  velocity,  in  direction,  or  in  all  three.     For  example,  a  con- 
tinuous rotation  with  uniform  velocity  may  transmit  continuous  rota- 
tion whose  velocity  is  greater  or  less,  uniform  or  variable,  in  the  same 
direction  or  the  reverse ;  it  may  transmit  rotation  intermittently  :  or 


MODIFICATION   OF   MOTION.  9 

the  follower  may  receive  a  reciprocating  motion  with  a  varying  velocity 
in  a  rectilinear  or  a  curvilinear  path.  In  an  elementary  combination, 
as  has  been  pointed  out,  the  path  of  each  piece  is  determined  by  its 
connection  with  the  frame- work  of  the  machine,  and  it  remains  for 
us  to  ascertain,  at  each  instant  of  the  action,  its  direction  and  velocity. 
If  this  be  done,  the  question  is  already  settled  as  to  whether  the  mo- 
tion has  been  changed  in  kind.  Hence  we  may  properly  say  that  the 
most  important  function  of  an  elementary  combination  is  to  modify 
motion  in  velocity  and  direction. 

27.  The  laws  which  govern  this  modification  are  determined  by  the 
comparative  movements  of  the  two  pieces.     It  is  apparent  on  reflection 
that,  at  every  instant  of  the  action,  or,  in  other  words,  for  every  possi- 
ble position  of  the  driver  with  respect  to  the  follower,  there  will  exist 
a  certain  definite  proportion  between  the  velocities,  and  an  equally 
definite  relation  between  the  directions,  of  their  motions ;  which  will 
depend  entirely  upon  the  pieces  themselves  and  the  manner  in  which 
they  act  upon  each  other,  and  cannot  be  affected  by  the  absolute 
directions  or  velocities. 

Consequently,  whatever  the  nature  of  the  combination,  the  analysis 
of  its  action  will  be  complete,  if  throughout  its  range  we  are  able  to 
determine,  as  between  the  driver  and  the  follower, 

1.  The  Velocity  Ratio. 

2.  The  Directional  Relation. 

28.  Now,  the  velocity  ratio  of  the  two  motions  may  remain  the  same 
during  the  entire  action,  or  it  may  vary ;  and  this  is  also  true  of  the 
directional  relation.     To  illustrate  :  If  two  circular  wheels  gear  with 
each  other,  turning  about  fixed  axes,  it  is  clear  that  the  velocity  ratio 
is  constant.     If  one  wheel  is  twice  as  large  as  the  other,  it  will  at  any 
instant  be  turning  half  as  fast,  whether  the  motions  be  uniform  or  not. 
And  so  of  the  relative  directions  of  the  rotations  ;  if  the  wheels  are  in 
external,  gear,  they  will  turn  in  opposite  directions,  if  they  are  in  in- 
ternal gear  they  will  turn  in  the  same  direction  :  but  the  directional 
relation,  whichever  it  may  be,  does  not  change.     If  two  elliptical 
wheels  engage,  the  directional  relation,  as  before,  is  constant  :  but  the 
velocity  ratio  will  change  as  the  radii  of  contact  vary.     In  the  case  of 
the  piston"  and  crank  of  a  steam  engine,  neither  the  velocity  ratio  nor 
the  directional  relation  is  constant ;  supposing  the  crank  to  turn  at  a 
uniform  rate  in  one  direction,  the  piston  travels  to  and  fro  with  a 
varying  speed. 

It  is  this  feature  of  constancy  or  the  reverse,  in  these  two  partic- 
ulars, which  distinguishes  the  actions  of  elementary  combinations 


10  BEPKESENTATIOK   OF  MOTION. 

from  each  other,  and  forms  the  true  basis  for  their  proper  classifi- 
cation. 

29.  Graphic  Representation  of  Motion. — It  has  already  been  suggested, 
that  the  action  of  a  combination  may  often  be  most  clearly  illustrated 
by  drawings  which  show  the  parts  in  their  proper  relative  positions  at 
convenient  phases  of  the  motion. 

But  the  motions  themselves  of  any  points  in  either  piece  also  admit 
of  perfect  graphic  representation  at  any  given  phase.  For  though 
the  path  of  a  moving  point  may  be  a  curve  of  any  kind,  yet,  as  the 
direction  of  a  curve  at  any  point  is  that  of  its  tangent  at  that  point, 
the  direction  of  the  motion  at  any  instant  may  be  indicated  by  that 
of  a  right  line^  And  the  velocity  being  an  abstract  number,  may  be 
properly  represented  by  the  length  of  that  line. 

30,  Geometrical  Method  of  Investigation. — Between  the  lines  thus 
representing  the'  motions  of  properly  selected  points,  and  other  lines 
closely  connected  with  the  moving  pieces,  definite  relations  may  usu- 
ally be  established,  in  such  a  manner  that,  by  means  of  diagrams  thus 
constructed,  the  velocity  ratio  and  the  directional  relation  may  be  as- 
certained in  the  particular  phase  represented,  and  the  law  governing 
the  modification  of  motion  throughout  the  action  deduced,  by  simple 
geometrical  reasoning. 

The  method  here  outlined  is  peculiarly  appropriate  to  this  subject, 
as  directly  leading  to,  if  not  directly  involving,  the  accurate  construc- 
tions of  the  movements  considered,  which  are  essential  in  practical 
operations  :  and  to  it  we  shall  adhere  throughout.  But  there  are 
some  general  principles  relating  to  motion  and  contact,  by  a  previous 
study  of  which  the  analysis  of  motion  as  modified  by  mechanical  de- 
vices will  be  much  facilitated  :  and  these  will  accordingly  receive  our 
attention  in  the  following  chapter. 


•HE       ar 


If  \ 

-{UNIVERSITY^ 


CHAPTER   II. 


COMPOSITION  AND  RESOLUTION"  OF  MOTION — INSTANTANEOUS  AXIS  OF 
ROTATION — MOTIONS  OF  TRANSLATION — COMPOSITION  OF  ROTATION 
AND  TRANSLATION — SLIDING,  ROLLING,  AND  MIXED  CONTRACT. 


Composition  and  Resolution  of  Motion. 


31,  Resultant. — If  a  material  point  receive  a  single  impulse  in  any 
direction,  it  will  move  in  that  direction  with  a  certain  velocity.     If  it 
receive  at  the  same  instant  two  impulses  in  different  directions,  it  will 
obey  both,  moving  in  an  intermediate  direction,  and  with  a  velocity 
different  from  that  due  to  either  impulse  alone.     Now  such  a  point 
may  receive  at  the  same  instant  any  number  of  impulses,  differing  in 
magnitude  and  direction.     But  the  point  can  move  only  in  one  direc- 
tion and  with  one  velocity;  this  actual  motion  is  called  the  Resultant; 
and  the  separate  motions,  which  the  different  impulses  taken  singly 
tended  to  give  it,  are  called  the  Components.     These  components  being 
represented  by* right  lines,  which  may  lie  in  the  same  or  in  different 
planes,  the  resultant  may  be  found  graphically  by  the  following  con- 
structions : 

32.  Parallelogram  of  Motions. — In  Fig.  1,  let  the  point  A  have  two 
component   motions,    represented  by  AB9   AC. 

These  two  lines  determine  a  plane,  in  which  the 
resultant  must  lie.     In  this  plane  draw  through 
B  a  parallel  to  AC,  and  through  C  a  parallel  to  A< 
AB.     These  parallels  intersect  at  D  ;   and  AD  FIO. 

is  the  resultant  sought. 

This  fundamental  proposition  may  be  thus  stated  :  If  two  compo- 
nent motions  be  represented  in  magnitude  and  direction  by  the  adjacent 
sides  of  a  parallelogram,  the  resultant  will  be  similarly  represented  by 
tJie  diagonal  passing  through  the  point  of  intersection. 


12  COMPOSITION   AND   KESOLUTION   OF  MOTION. 

83.  Composition  of  Motions. — This  process  of  finding  the  resultant  of 
simultaneous  independent  motions  is  called  Composition :  and  any 
number  of  them  may  be  compounded  by  repeating  it.  Thus  in  Fig. 

2,  let  there  be  three  components,  AB, 
AC,  AE,  all  lying  in  the  plane  of  the 
paper.  We  first  compound  any  two  of 
them,  say  AB  and  A  C,  as  in  the  figure, 
giving  a  resultant  AD,  which  is  next 
compounded  with  AE,  by  which  finally 
we  find  AF,  the  required  resultant.  The 
process  is  the  same  for  any  number  of 
components,  and  it  makes  no  difference  in  what  order  they  are  taken. 

34.  Parallelopipedon  of  Motions. — The  above  holds  true  whether  the 
components  all  lie  in  the  same  plane  or  not.     In  Fig.  3,  let  AB,  AC, 

AE,  be  three  components,  the  first  two 
lying  in  the  plane  of  the  paper,  while 
the  last  does  not.  Proceeding  as  before, 
AD,  the  resultant  of  AB  and  AC,  will 
lie  in  the  plane  of  the  paper  also  ;  but 

AF,  the  final  resultant,  found  by  com-  FIG.  3. 
pounding  AD  with  AE,  will  lie  in  a  different  plane,  determined  *by 
those  two  lines.    Had  there  been  a  greater  number  of  components,  we 
should   have  continued  in  a  similar  manner,  compounding  AF  \f\i\\. 
one  of  them,  their  resultant  with  another  component,  and  so  on.     But 
the  figure  sufficiently  illustrates  not  only  the  process,  but  also  another 
fundamental  proposition,  relating  to  three  components  not  in  one 
plane,  which  may  be  thus  expressed  : 

If  three  component  motions  be  represented  by  tlie  three  adjacent  edges 
of  a  parallelopipedon,  the  resultant  will  be  represented  in  magnitude 
and  direction  by  the  body  diagonal  which  passes  through  the  point  of 
intersection. 

35.  Resolution  of  Motion. — This  is  the  inverse  process  to  that  above 
explained.     It  is  obvious  that  if  two  or  more  independent  motions  can 
be  compounded  into  a  single  equivalent  motion,  that  resultant  can  be 
again  separated,   or  resolved,  into  its  components.     This  would  be 

#0  ^..9  just  as  true  if  the  resultant  .thus  found  had 

been  originally  given  as  an  independent  sin- 
gle motion.  Consequently  any  motion  may 
be  resolved  into  two  components,  either  of 
these  into  two  others,  and  so  on  ad  libitum. 

And  these  components  may  be  given  any  direction  at  pleasure.     In 
Fig.  4  let  AB  represent  the  given  motion  ;  through  A  draw  A  C  in  any 


COMPOSITION  AND   KESOLUTION   OF  MOTION.  13 

direction,  and  through  B  draw  BC,  cutting  A  C  in  any  point  C.  Then 
completing  the  parallelogram,  it  is  evident  that  AB  is  the  resultant 
of  the  two  motions  represented  by  A  C,  AD.  Of  these  the  direction 
of  the  first  was  assigned ;  that  of  the  other  is  not  wholly  arbitrary, 
since  it  must  lie  in  the  same  plane  with  AB  and  A  C.  But  AD  may 
be  again  resolved,  the  direction  of  one  component  being  assumed  as 
before :  and  this  may  be  repeated  until  components  of  the  original 
motion  have  been  found,  in  as  many  different  directions  in  space  as 
may  be  desired. 

36.  From  the  above  it  will  be  perceived  that  if  the  given  motion 
and  the  required  components  all  lie  in  one  plane,  whatever  the  num- 
ber of  the  latter,  the  resolution  can  be  exactly  effected.     If  they  do 
not,  there  may  be  one  component  in  addition  to  those  whose  directions 
are  assigned  :  should  these  all  lie  in  one  plane  which  does  not  contain 
the  given  motion,  this  must  obviously  be  the  case. 

But  a  motion  can  be  exactly  resolved  into  three  components  not  in 
one  plane,  provided  that  it  does  not  lie  in  the  plane  determined  by 
either  two. 

Thus  in  Fig.  3,  let  AF  be  the  given  motion,  and  Ax,  Ay,  Az,  the 
directions  of  the  three  components.  Now  these  three  lines  last  men- 
tioned determine  three  planes ;  and  by  passing  through  F  three 
other  planes  respectively  parallel  to  these,  a  parallelopipedon  is  con- 
structed, of  which  AF  is  the  body  diagonal,  and  the  edges  pass- 
ing through  A  are  therefore  the  required  components. 

37.  Normal  and  Tangential  Components. — It  is  often  required  to  resolve 
a  motion  into  two  components,  of  which  one  shall  be  perpendicular,  the 
other  parallel  to  a  given  plane  :  usually  a  plane  tangent  to  the  surface 
of  some  moving  piece  under  consideration.     In  this  case  the  former  is 
called  the  normal,  and  the  latter  the  tan- 
gential, component.    They  are  easily  found 

as  in  Fig.  5.  Let  A  be  the  moving  point, 
AB  representing  its  motion  :  draw  through 
A  the  plane  MN  parallel  to  the  given 
plane  ;  upon  this  let  fall  the  perpendicular 
BC,  and  draw  A  C.  Completing  the  paral- 
lelogram CD,  it  will  be  perceived  that  the 

tangential  component  is  AC,  the  orthographic  projection  of  AB  upon 
the  plane  MN,  and  that  AD,  the  normal  component,  is  equal  and 
parallel  to  BC,  the  projecting  perpendicular  of  the  point  B. 

38.  Motions  of  Connected  Points.— If  two  points  be  so  connected  that 
the  distance  between  them  is  invariable,  they  may  be  supposed  to  be 
connected  by  a  right  line.     The  motions  of  the  points  at  any  instant 


14 


INSTANTANEOUS   AXIS   OF  ROTATION. 


may  be  represented  by  right  lines,  which  may  be  in  the  same  or  in 
different  planes.  Each  of  these  may  be  resolved  into  two  components, 

one  of  which  .s  in  the  direction  of 
the  connecting  line,  the  other  per- 
pendicular to  it.  And  the  simulta- 
neous motions  are  subject  to  this  con- 
dition, that  the  components  along  the 
line,  thus  found,  must  have  the  same 
magnitude  and  direction.  Thus  in 
FlG-  6-  Fig.  6,  the  components  A  C,  BF,  along 

the  line  AB,  must  be  equal  and  in  the  same  direction  ;  otherwise 
the  distance  AB  would  change,  which  is  contrary  to  the  hypothesis. 
These  components  are  at  once  found  by  drawing  through  E  and  D9 
planes  perpendicular  to  AB,  cutting  it  in  C  and  F. 


The  Instantaneous  Axis  of  Rotation. 

39.  A  right  line,  moving  in  any  manner  in  space,  may  at  any  given 
instant  be  regarded  as  revolving  about  some  other  right  line  more  or 
less  remote.     The  latter  may,  from  instant  to  instant,  change  its  posi- 
tion not  only  in  space,  but  in  relation  to  the  line  whose  motion  is 
under  consideration,  with  reference  to  which  it  is  therefore  called  the 
Instantaneous  Axis. 

40.  If  the  motions  of  two  given  points  in  a  right  line  are  known, 
the  motion  of  the  whole  line  is  fully  determined.     Consequently,  if  it 
can  be  shown  that  the  actual  motions  of  these  two  points,  at  any  one 
instant,  are  the  same  as  though  they  were  revolving  with  the  same 
angular  velocity  and  in  the  same  direction  about  any  axis,  the  motion 
of  the  line  is  at  that  instant  equivalent  to  one  of  revolution  about  the 
same  axis. 

First,  considering  the  motion  of  a  single  point  in  space  :  whatever 
the  path,  the  motion  is,  at  any  given  instant,  equivalent  to  one  of 
revolution  about  any  right  line  in  the  plane  normal  to  the  path  at  the 
position  occupied  at  that  instant  by  the  moving  point.  Thus  in  Fig. 
7,  let  P  be  the  moving  point,  PT  the  tan- 
gent to  its  path,  and  MN the  normal  plane. 
In  this  plane  draw  any  right  line  AB,  and 
PC  perpendicular  to  it.  Then  if  P  be  sup- 
posed to  revolve  about  AB  as  an  axis,  it 
will  describe  a  circle  whose  plane,  being  FlG.  7. 

perpendicular  to  MN,  will  contain  the  line  PT:  and  this  line  being 
perpendicular  to  the  radius  CP  at  its  extremity,  will  be  tangent  to 


INSTANTANEOUS   AXIS   OF   ROTATION. 


15 


the  circle  thus  described.  Similarly  the  actual  motion  of  P  may  be 
proved  equivalent  to  one  of  revolution  about  HL,  or  any  other  right 
line  lying  in  the  normal  plane  MN. 

The  angular  velocity  will  depend  upon  the  distance  of  the  moving 
point  from  the  assumed  axis,  and  is  conveniently  measured  by  divid- 
ing the  given  linear  velocity  by  that  distance,  since  in  general  we 
have  (11) 


ang.  vel. — 


lin.  vel. 
rad. 


Thus  if  in  the  figure,  PT  represent  the  linear  velocity  of  P  at  the 
given  instant,  we  shall  have  for  the  equivalent  rotations, 


and 


ang.  vel.  about  A B  —  z 


ang.  vel.  about  HL  = 


PT 
PD 


41.  Now,  if  the  motions  of  two  given  points  on  a  right  line  be 
known,  the  above  reasoning  applies  to  each.    Consequently,  if  we  draw 
two  planes,  respectively  normal  to  the  paths  at  the  positions  simulta- 
neously occupied  by  the  moving  points,  the  intersection  of  these  planes 
will  be  the  only  right  line  about  which  both  points  can  at  the  given 
instant  be  regarded  as  revolving. 

But  we  have  already  seen  that  the 
motions  of  two  connected  points 
are  subject  to  another  condition 
(38),  and  it  remains  to  be  shown 
that  these  equivalent  revolutions 
will  have  the  same  direction  and 
the  same  angular  velocity  when 
that  condition  is  satisfied. 

42,  We  will  first  consider  the  case 
in  which  all  the  motions  of  the  line 
upon  which   the  given  points  are 
situated,  are  confined  to  one  plane. 
In  Fig.  8,  let  AB  be  the  moving 
line,  A  and  B  the  given  points.   Let 
AD  represent  the  motion  of  A  in 

magnitude  and  direation,  and  sup-  FIO.  9. 

pose  B  to  move  in  the  direction  BE\AD  may  be  resolved  into  the 


16  INSTANTANEOUS   AXIS   OF   ROTATION. 

two  components,  A  C  in  the  direction  of  AB,  and  AF  perpendicular 
to  it.  The  motion  of  B  must  have  a  component  BG  along  AB, 
equal  to  AC  and  in  the  same  direction.  Having  set  this  off,  draw 
at  G  a  perpendicular  to  AB,  cutting  BE  at  E.  This  determines  the 
linear  velocity,  BE,  of  the  point  B,  and  completing  the  parallelo- 
gram, this  is  seen  to  be  the  resultant  of  the  two  components  BG,  BH. 
Suppose  all  these  lines  to  lie  in  a  horizontal  plane  parallel  to  the  paper  ; 
then  AO  perpendicular  to  AD,  and  BO  perpendicular  to  BE,  will  be 
the  traces  of  the  two  normal  planes,  and  .their  intersection  0  will 
represent  a  vertical  line.  Now  draw  OL  perpendicular  to  AB,  then 
the  triangles  ACD,  ALO,  are  similar,  and  also  the  triangles  EGB} 
BLO  ;  whence 


and 


.AC  _ 
OL  ~  OA 


OL  ~-  OB 


AD       BE 


~  OB 


But  since  AD,  BE,  are  the  linear  velocities  of  A  and  B,  the  frac- 

tions -—  and  —  —  are  the  angular  velocities  of  the  revolutions  of  those 
C/_/l  (J  Jj 

points  about  0  ;  which  are  thus  proved  to  be  equal,  and  their  direc- 
tions are  the  same  by  construction. 

43.  The  same  pairs  of  similar  triangles  will  therefore  give 


A~L  "  UA  ~  OB  ~  IL 


but  CD  =  AF,  and  GE  =  BH  ; 
whence 


AF      BH 


Therefore  the  right  linS  FH  must  pass  through  L,  the  foot  of  the 
perpendicular  from  0  upon  AB.  This  fact  is  important,  for  the 
angle  between  AO  and  BO  may  be  so  acute  as  to  render  it  difficult 
to  determine  the  exact  point  of  intersection.  In  that  case,  if  the  rel- 
ative velocities  are  known,  we  have  only  to  find  the  components  per- 
pendicular to  A»B,  which  may  be  laid  off  upon  any  scale,  so  that  FH 
need  not  cut  AB  acutely  ;  and  the  point  L  can  thus  be  found  v/ith 
precision. 


INSTANTANEOUS  AXIS   OF   KOTATION.  17 

44.  The  general  case  is  that  in  which  the  motions  of  the  two  given 
points  are  in  different  planes ;  in  illustration  of  which,  Fig.  8,  still 
regarded  as  a  horizontal  projection,  is  to  be  studied  in  connection  with 
the  vertical  projection  given  in  Fig.  9.     AD  and  BR  are  parallel 
to  the   horizontal  plane,   to  which  AB  is    inclined,    though  par- 
allel to   the  vertical   plane.       In  Fig.    9,    therefore,    AB  appears 
as    A'B',    and  AD    is  foreshortened    into   A'D'  :    and    the    actual 
component  A'M  along  the  line  is  found  as  in  Fig.   6,  by  passing 
through  D'  a  plane  perpendicular  to  A'B'.      Then   making  B'N, 
the  corresponding  component   of   the  motion  of  B,  equal  to  A'M, 
and  drawing  a  plane  through  ^perpendicular  to  A'B',  we  have  B'E' 
the  vertical  projection  of  the  resultant ;  from  this  the  horizontal  pro- 
jection BE,  which  is  seen  in  its  true  length,  is  determined,  and  is 
evidently  of  precisely  the  same  length  as  in  the  previous  case.     Now 
AD  and  BE  may  as  before  be  resolved  into  the  components  AF,  AC, 
and  BG,  BH,  lying  in  horizontal  planes.     The  preceding  argument 
depends  upon  these  only,  and  accordingly  applies  without  change. 

45.  In  general,   then,  the  instanta- 
neous axis  of  a  moving  right  line  may 
be  found  if  the  directions  only  of  the 
simultaneous  motions    of    two  points 
upon  it  are  given  :  but  this  may  not  be 
possible  if  those  directions  are  parallel. 
Thus  in  Figs.  10  and  11,  let  A  C  and  BE, 
the  motions  of  the  points  A  and  B  at  the 
given  instant,  be  parallel  to  each  other 
and  to  the  paper,  and  perpendicular  to 
the  moving  line  AB.     In  this  case  the 
two  normal  planes  coincide  ;  yet  there 
is  an  instantaneous   axis,  in   order  to 

locate  which  we  must  know  also  the  FlG> 

relative  velocities  at  least  of  the  two  points.  Setting  these  off  as  AC, 
BE,  of  their  proper  proportionate  lengths  by  any  scale,  CE  will  cut 
AB  or  its  prolongation  in  D.  If  we  conceive  a  right  line  perpendic- 
ular to  the  paper  at  D,  it  will  lie  in  the  normal  planes,  and  the  motion 
of  AB  is  evidently  equivalent  to  one  of  rotation  about  it.  And  this  will 
as  obviously  be  true,  whatever  may  be  the  inclination  of  the  moving 
line  to  the  plane  of  the  paper.  Any  line  in  the  normal  plane,  there- 
fore, which  passes  through  D,  maybe  taken  as  the  instantaneous  axis. 

46.  Now,  in  Figs.  10  and  11,  let  AB,  AC,  be  regarded  as  parallel 
to  the  plane  of  the  paper,  while  BE  is  the  projection  of  a  line  inclined 
to  that  plane,  though  perpendicular  to  AB.     Then  the  planes  normal 

2 


18 


MOTION   OP  TRANSLATION. 


FIG.  12. 


to  A  0  and  BE  will  intersect  in  the  line  A B  itself ;  which  is  then  its 
own  instantaneous  axis,  about  which  we  may  for  the  sake  of  consist- 
ency say  that  it  is  revolving  with  infinite  velocity.  It  is,  certainly, 
the  only  line  about  which  it  can  be  said  to  have  a  motion  of  rotation 
only  ;  it  does  not  follow,  however,  that  it  will  remain  stationary  in 
space,  which  it  will  not  do  :  but,  as  will  be  shown  subsequently,  its 
actual  motion  will  be  helical. 

47.  Motion  of  Translation. — From  consideration  of  Fig.  11  it  will  be 
seen  that  the  more  nearly  equal  A  C  and  BE  are,  the  more  remote  will 
be  the  intersection  D  :  and  that  if  they  be  exactly  equal,  the  instanta- 
neous axis  will  be  at  an  infinite  distance.     Since  in  that  case  all  the 
points  of  AB  are  at  the  instant  traveling  in  the  same  direction  with 
the  same  velocity,  the  consecutive  position  of  the  moving  line  will  be 

_  parallel  to  the  present  one.     This 

will  occur  whenever  the  simulta- 
neous motions  of  the  two  given 
points  are  in  parallel  lines  and  in 
the  same  direction,  whether  per- 
pendicular to  the  moving  line  or 
not.  Thus  in  Fig.  12,  the  normal 
planes  AM,  BN,  are  parallel,  al- 
though the  motions  of  A  and  B  are  inclined  to  the  line  AB.  It  is 
evident  that  in  this  case  A  C  and  BE,  the  actual  linear  velocities  of 
the  two  points,  must  be  equal,  since  (38)  the  components  A H  and 
BG,  along  the  line,  must  be  equal  and  lie  in  the  same  direction. 
The  motion  of  AB  in  Fig.  12  may  then  be  regarded  as  one  of  rota- 
tion about  an  infinitely  remote  line  parallel  to  BN,  and  lying  in  a 
plane  parallel  to  AB :  if  AC,  BE,  are  made  perpendicular  to  AB, 
the  instantaneous  axis  will  be  infinitely  remote  and  parallel  to  the 
moving  line. 

48.  The  motion  of  the  line,  at  any  instant  when  it  is  thus  revolving 
about  an  axis  at  an  infinite  distance,  is  called  Translation.     It  may  be 
continuous  ;  and  not  only  a  right  line,  but  any  geometrical  magnitude, 
is  said  to  have  a  motion  of  translation,  when  all  its  points  move  with 
the  same  velocity  and  in  the  same  direction,  in  equal  and  similar 
paths.     Under  these  circumstances,  evidently,    all  the  tangents  to 
these  paths,  at  the  positions  simultaneously  occupied  by  the  moving 
points,  will  be  parallel  :  and  the  right  line  joining  any  two  of  these 
points  will  remain  parallel  to  itself  during  the  whole  movement.     If 
the  paths  are  rectilinear,  the  motion  is  called  straight  translation,  or, 
more  commonly,  sliding  :  if  they  are  curvilinear,  the  motion  is  called 
circular,  elliptical,  or  helical,  translation  as  the  case  may  be,  the  name 


COMPOSITION   OF   ROTATION   AND   TRANSLA 

depending  on  the  form  of  the  curve.     A  familiar  instam 
translation  is  afforded  by  the  coupling-rod  of  a  locomotive,  shown  in 
Fig.  13  :  the  points  A  and  B  move  as  indicated  by  the  arrows,  in  the 
circles  whose  centres  are  D  and  E, 
with  the  same  velocity.     And  any 
other  point   C,  in  a  rigid  exten- 
sion of  the  bar  AB,  will  move 
similarly  in  the  equal  circle  whose 
centre  is  F. 

49,  In  relation  to  a  single  right 
line,  it  is  to  be  noted  that  a  mo- 
tion of  circular  translation,  if  the 

planes  of  the  circles  be  perpendicular  to  the  moving  line,  is  identical 
with  one  of  revolution  about  the  axis  of  the  cylinder  thus  generated. 
Thus,  in  Fig.  13,  the  point  C  may  be  considered  as  a  line  perpendicu- 
lar to  the  paper ;  and  its  motion  is  the  same  as  though  it  revolved 
about  a  parallel  axis  passing  through  the  point  F. 

It  is  also  to  be  observed  that  the  motion  of  a  single  right  line  of  a 
rigid  body  does  not  determine  the  motion  of  the  whole  body,  which  if 
not  otherwise  controlled  would  be  free  to  rotate  about  that  line.  But 
if  the  motions  of  two  right  lines  of  the  body  be  given,  those  of  all  its 
other  points  are  thereby  fully  determined. 


Composition  of  Rotation  and  Translation. 

50.  Now,  just  as  a  simple  motion  in  a  right  line  can  be  resolved  into 
two  others,  so  a  simple  rotation  about  a  fixed  axis  may  be  regarded  as 
•a  resultant  of  two  other  motions,  viz.,  a  rotation  about  another  axis 
parallel  to  and  revolving  around  the  first,  and  a  translation  in  which 
the  paths  are  circles  whose  radii  are  equal  to  the  distance  between 
these  axes  ;  the  angular  velocities  and  the  di- 
rections of  both  components  being  the  same 
as  those  of  the  rotation  which  was  to  be  re- 
solved. 

Thus  in  Fig.  14,  let  C  represent  a  fixed  axis 
perpendicular  to  the  paper,  and  let  MAN  re- 
volve about  this  axis  into  the  position  8BT. 
We  may  also  let  A  represent  a  right  line  per- 
pendicular to  the  paper,  about  which  MN  may  rotate  into  the  posi- 
tion UV,  afterward  moving  by  circular  translation  to  the  position 
ST.  Or,  MN  may  move  by  circular  translation  into  the  position  PR, 
afterward  rotating  about  B  into  the  position  ST ':  the  path  of  A  being 


20  RESOLUTION   OF   CIRCULAR  TRANSLATION. 

in  all  cases  the  arc  AB.  On  either  supposition  the  result  is  the  same 
as  if  both  motions  progress  simultaneously  and  uniformly,  which  pro- 
duces the  original  revolution  about  C :  and  the  angles  J/^T*y  RBT, 

and  A  CB9  are  equal  by  construc- 
tion. In  Fig.  15,  the  traveling 
axis  passes  through  N;  now  let 
the  points  M  and  A,  during  trans- 
lation from  position  MN to  position 
PT,  describe  circles  whose  radii  are 
equal  to  CN.  Then  if  in  the  same 
time  MN  rotate  about  N,  in  the 
same  direction,  through  an  angle 
PTS  =  NCT,  it  will  come  into  the  same  position,  ST,  as  in  Fig.  14. 
In  short,  any  axis  parallel  to  the  fixed  one  may  be  assumed  as  that  of 
the  component  rotation,  and  thus  the  revolution  around  C  may  be  re- 
solved in  an  infinite  number  of  ways. 

51.  On  the  other  hand,  a  motion  of  circular  translation  may  be  con- 
sidered as  a  resultant.     For  (49)  any 

line  perpendicular  to  the  planes  of  the 
translation,  and  partaking  of  that  mo- 
tion, may  be  regarded  as  revolving 
about  a  fixed  axis.  Assuming  such  M 
line  as  a  traveling  axis  of  rotation, 
the  component  motions  arc, first:  a 
revolution  about  that  fixed  axis,  the  ° c 

direction  and  angular  velocity  being 

the  same  as  those  of  the  translation;  and,  second,  a  rotation  in  the 
opposite  direction,  but  with  the  same  angular  velocity,  about  the  as- 
sumed traveling  axis.  Thus  in  Fig.  16,  if  MAN  revolve  about  the 
fixed  axis  C,  until  A  reaches  B,  and  this  be  its  only  motion,  it  will 
have  the  position  SBT.  If  in  the  same  time  it  also  rotates  in  the 
opposite  direction  about  A,  as  shown  by  the  arrows,  through  an  angle 
TBR,  equal  to  A  CB,  its  new  position  will  be  PBR,  parallel  to  the 
original  position  MAN;  and  the  resultant  motion  will  be  one  of  cir- 
cular translation,  in  which  M,  N,  and  all  the  moving  points,  describe 
circles  whose  radii  are  equal  to  A  C. 

52.  If   any  geometrical  magnitude  have  a  motion  of   rectilinear 
translation  parallel  to  an  axis  about  which  it  is  revolving,  the  result- 
ant is  a  helical  motion  about  that  axis.     But  a  motion  may  be  helical 
with  respect  to  one  axis,  and  yet  simply  rotatory  with  respect  to  an- 
other.    For  instance,  the  rectilinear  generatrix  of  an  oblique  helicoid 
has  a  continuous  helical  motion  about  the  fixed  axis  of  the  surface ; 


KESULTAKT   MOTION   OF   A   KIGID   BODY. 


but  in  any  given  position  of  the  moving  line,  its  motion  is  equivalent 
to  one  of  rotation  only  about  an  instantaneous  axis,  determined  (41) 
by  the  intersection  of  planes  normal  to  the  paths  of  any  two  of  its 
points  :  and  either  may  be  regarded  as  the  actual  motion  of  the  gene- 
ratrix at  that  instant,  the  latter  being  the  less  complex.  In  the  case 
of  the  right  helicoid,  the  intersection  of  these  normal  planes  is  the 
generatrix  itself  ;  and  the  motion  of  that  line  in  space  cannot  be  re- 
duced to  any  simpler  form  than  that  of  a  helical  one  about  an  axis, 
which,  in  this  special  case,  is  at  every  instant  the  same,  being  the 
fixed  axis  of  the  surface. 

53.  But,  in  general,  if  the  motions  of  two  connected  points  are 
perpendicular  to  the  right  line  joining  them, 
and  are  not  parallel  to  each  other,  the  mo- 
tion of  that  line  will  be  a  helical  one  about 
an  instantaneous  axis :  which  may  be  de- 
termined, if  the  relative  velocities  are  given, 
in  the  following  manner  : 

In  Figs.  17  and  18,  the  moving  line  AB 
is  perpendicular  to  the  vertical  plane,  and 
B'  is  its  vertical  projection.  The  motions 
of  the  points  A  and  B  are  seen  in  their  true 
lengths  and  directions  in  the  vertical  pro- 
jection as  B'C',  B'E'  (for  convenience  so  • 
drawn  that  C'E'  is  horizontal),  and  their 
horizontal  projections  are  A  C,  BE.  Through 
AB  pass  a  plane  parallel  to  C'E',  and  resolve  these  motions  normally 
and  tangentially  with  reference  to  this  plane.  The  tangential  com- 
ponents are  B  G,  B'F,  horizontally  projected  in  AC,  BE.  Draw  EC, 

and  produce  it  if  necessary  to  cut  BA  or 
its  prolongation  in  D,  whose  vertical  pro- 
jection is  B'.  Then  the  motion  due  to 
these  components,  which  are  horizontal 
and  parallel,  will  be  (45)  a  rotation  about 
any  axis  passing  through  D,  and  lying  in 
the  vertical  pla*ne  which  contains  AB. 
The  normal  components  are  vertical  and 
equal,  the  true  length,  B'H,  being  seen  in 
the  vertical  projection.  The  motion  of 
AB  due  to  these  components,  therefore 
is  one  of  vertical  translation  ;  and  the 
actual  motion  in  space  is  a  helical  one  about  a  vertical  axis  LL  pass- 
ing through  D. 


FIG.  17. 


B' 


r. 

FIG.  18. 


22  RESULTANT  MOTION   OF  A   RIGID   BODY. 

54,  Let  the  points  of  a  material  right  line  be  so  connected  with 
those  of  a  rigid  body  that  the  distances  between  them  cannot  change. 
We  may  then  suppose  that  line  to  move,  and  the  body  primarily  to 
move  in  the  same  manner.     If  now  the  body  also  rotate  about  the 
line  as  an  axis,  then  these  two  motions  may  be  compounded  :  and  (49) 
if  it  can  be  shown  that  any  two  lines  of  the  rigid  body  are  at  the  same 
instant  moving  in  a  rotatory  or  helical  manner  about  any  axis,  the  whole 
body  may  at  that  instant  be  regarded  as  doing  the  same.     Evidently 
the  moving  line  itself  may  be  taken  as  one  of  these  two,  and  any  line 
intersecting  it  as  the  other. 

The  problem  of  determining  the  resultant  presents  several  different 
cases,  depending  upon  the  nature  of  the  motions  primarily  assigned 
to  the  rigid  body  and  the  line  with  which  it  is  connected  ;  and  these 
will  be  considered  separately. 

55.  I.  Let  the  primary  motion  be  one  of  revolution  about  an  axis 
parallel  to  the  line. 

The  resultant  will  then  be  a  rotation  of  the  body  about  an  instan- 
taneous axis,  which  will  be  parallel  to  the  other  two  axes,  and  lie  in 
the  same  plane. 

In  Fig.  19,  let  C  represent  the  moving  line,  P  the  axis  about  which 

it  is  revolving,  both  being  perpen- 
dicular to  the  paper ;  and  let  CD, 
perpendicular  to  CP,  and  parallel  to 
the  paper,  represent  the  linear  veloc- 
ity of  any  point  of  the  moving  line 
in  its  revolution  around  P.  Join 
any  point  of  this  line  with  any  point 
A  of  the  body  :  then  CA  represents 
a  line  which  may  or  may  not  be  in- 
clined to  the  paper.  Let  AE,  per- 
pendicular to  CA,  be  the  linear  velocity  of  A  in  its  rotation  about  Cj 
the  motion  of  A  due  to  the  revolution  about  P  will  be  AB,  perpen- 
dicular to  AP,  the  magnitude  being  determined  by  making  the  angle 
APB  equal  to  the  angle  CPD.  Both  AE  and  AB,  and  therefore 
their  resultant  AF,  the  actual  motion  of  A,  are  parallel  to  the  paper. 
Consequently  the  normal  planes  NN,  MM,  of  which  the  latter  con- 
tains the  axes  C  and  P,  intersect  at  0  in  a  line  perpendicular  to  the 
paper,  which  is  the  instantaneous  axis. 

The  location  of  this  axis  may  be  found  otherwise,  thus  :  regarding 
CP  as  a  line  of  the  rigid  body  (which  may  be  inclined  or  parallel  to 
the  paper),  the  rotation  of  P  around  tf  will  be  represented  by  PG, 
the  angle  PCG  being  made  equal  to  the  angle  ACE :  then  (45)  GD 
will  cut  CP  in  0. 


COMPOSITION   AND   KESOLUTION   OF   MOTION. 


56.  II.  Let  the  primary  motion  be  one  of  translation,  in  which  the 
paths  are  in  planes  perpendicular  to  the  line. 

In  this  case  also  the  resultant  will  be  a  rotation  about  an  instanta- 
neous axis  parallel  to  the  line. 

This  will  be  apparent  from  inspection  of  Fig.  20,  the  construction 
differing  from  the  preceding  only  in 
this ;  that  AB  is  parallel  and  equal  to 
CD,  the  motion  of  C  in  its  path  RS, 
whose  radius  of  curvature  at  C  is  for 
the  purpose  of  comparison  made  equal 
to  CPof  Fig.  19  :  also  CD  and  AE are 
of  the  same  lengths  in  both  diagrams, 
in  which  therefore  themo  tions  of  C  at  FIG.  20- 

the  given  instant  are  identical,  and  the  rotations  about  the  moving 
line  are  alike. 

57.  III.  Let  the  primary  motion  be  one  of  revolution  about  an 
axis  which  intersects  the  line. 

•  The  resultant  will  then  be  a  rotation  about  an  instantaneous  axis 
lying  in  the  plane  of  the  other  two,  and  passing  through  their  point 
of  intersection. 

In  Fig.  21,  let  CO  be  the  line,  OA  the  axis,  both  in  the  vertical 
plane.  Draw  CA  at  pleasure,  and  suppose 
it  a  line  of  the  rigid  body.  Then  the  mo- 
tion of  A  is  due  only  to  the  rotation  about 
CO,  that  of  C  is  due  only  to  the  primary 
revolution  about  OA,  and  both  are  perpen- 
dicular to  the  vertical  plane.  Let  these  mo- 
tions be  represented  by  A  'E,  C'D,  in  the  hor- 
izontal projection  ;  and  draw  DE,  cutting 
C'A'  in  P',  whose  vertical  projection  is  P. 
Then  (45)  CA  may  be  regarded  as  rotating 
about  any  axis  lying  in  the  vertical  plane  and 
passing  through  P,  and  CO  as  rotating 

'A 


about     any 

axis  in  the 

same  plane, 

and  passing  through  0.  Consequent- 
ly OP  is  the  instantaneous  axis  of 
both  CO  and  CA,  and  therefore  of 
the  whole  body. 

58.  Draw  IPK  perpendicular  to 
OP  and  consider  it  another  line  of  FIG.  22. 


FIG.  21. 


24 


COMPOSITION   AND   RESOLUTION   OF  MOTION. 


the  rigid  body  :  then  -in  Fig.  22,  draw  KG  perpendicular  and   PL 
parallel  to  0(7,  also  IH  perpendicular  and  PM  parallel  to  OA,  and 
join  PG. 
Now  let 


V  —  ang.  vel.  of  K      around  00, 

y>=    «        «     «  i  «        OA)  i    Then 


V    = 


/or  K 


OP. 


v 
V 


Iff 


By  construction,  OPK,  OGK  are  right  angles,  therefore  a  circle 
will    go    round    OGPK,  whose    diameter  =   OK  '  ; 
in  which  KOP  =  KGP,  standing  on  same  arc  'PK, 
and  GKP  =    GOP,         "        "     "       "  GP  ; 

but  OLP  =  GOP,  by  reason  of  parallels  PL,  OG: 

.  •  .  triangles  GPK,  OPL,  are  similar. 

whence 

V      PK  _PL       OM 

v  ~  KG  -  OP  ~  OP  : 
similarly 


.   - 
v  ~  IH 


OP* 


That  is  to  say  :  that  if  we  lay  off  upon 
the  axes  the  distances  OM,  OL,  propor- 
tional to  the  angular  velocities  of  the  rota- 
tion and  the  revolution  about  those  axes 
respectively,  and  complete  the  parallelo- 
gram ML,  its  diagonal  OP  will  lie  in  the 
instantaneous  axis,  and  be  proportional  to 
the  angular  velocity  of  the  resultant  rota- 

tion about  it. 
In  Figs.  21 
and  22,  the 
direction  i  n 
which  the  rigid 
body  rotates 
about  the 
traveling  axis, 
and  that  in 
which  the 
traveling  axis 
revolves  around 

FIG.  23. 


KESULTAKT   MOTIOH   OF  A   EIGID   BODY. 


the  fixed  one,  are  the  same.  But  the  above  reasoning  applies  equally 
well  if  they  are  opposite  ;  and  Figs.  23  and  24  show  the  modifications 
in  the  diagrams  due  to  this  change  in  the  conditions.  In  regard  to 
which  it  will  be  noted  that  in  both  cases  OH  is  set  off  in  the  same 
direction  from  0  ;  but  OL  is  measured  in  one  direction  if  the  rota- 
tion and  the  revolution  are  alike,  and  in  the  other  if  they  are  unlike. 

59.  IV.  Let  the  body  and  the  line  primarily  revolve  about  another 
line  in  a  different  plane. 

In  this  case  the  resultant  will  be  a  helical  motion  of  the  whole  body 
about  an  instantaneous  axis  lying  in  a 
plane   parallel   to   the   other  two   lines, 
and  intersecting  their  common  perpen- 
dicular. 

In  Fig.  25,  let  OA  be  the  fixed  axis, 
OC  the  moving  one,  both  parallel  to  the 
vertical  plane  ;  0  being  the  vertical 
and  DG  the  horizontal  projection  of 
their  common  perpendicular.  Consider- 
ing DG  as  a  line  of  the  rigid  body,  let 
the  motion  of  D  be  represented  by  OE 
perpendicular  to  OA,  that  of  G  by  OS 
perpendicular  to  00;  the  axes  being 
for  convenience  so  placed  that  BE  is  horizontal. 

Eesolve  these  motions  normally  and  tangentially  with  reference  to 
a  horizontal  plane  through  0  :  the  normal  components  will  be  verti- 
cally projected  in  OR,  the  tangential  ones  in  ON,  OM,  of  which  the 
horizontal  projections  are  DF,  GU:  therefore  UP  is  the  horizontal 
projection  of  a  line  joining  the  points  M  and  N. 

Produce  MB  to  cut  OA  in  A,  also  NE  to  cut  OC  in  C;  then 
A  C  will  be  parallel  to  MN.  For  the  triangles  OMA,  ONE,  are 
similar  :  so  are  also  the  pair  OMB,  ONC:  and  BM  =  EN.  Hence 
we  have 

AM.      OM       ..CNOM         A  „        ^ 
ON  ~  EN*  and  OJf=        '  •*•  AM  = 


and  UF  is  also  the  horizontal  projection  of  A  C. 

60.  Moreover,  UF  is  the  trace  on  a  horizontal  plane  through  A,  of 
a  plane  passing  through  the  point  C  and  the  axis  0  A  .  The  actual 
motion  of  C  about  that  axis,  is  perpendicular  to  this  plane  ;  therefore 
its  horizontal  projection  F8  must  be  perpendicular  to  UF.  Its  verti- 
cal projection  CH  must  (see  Fig.  9)  be  parallel  and  equal  to  OE  ; 
whence  the  magnitude  of  F8  is  determined.  With  reference  to  the 


26  RESULTANT   MOTION   OF   A   RIGID   BODY. 

vertical  plane,  this  motion  may  be  resolved  into  a  normal  component 
FQ,  vertically  projected  in  C,  and  a  tangential  one  of  which  the  pro- 
jections are  FL,  CH.  The  latter  component  being  parallel  and  equal 
to  OE,  may  be  resolved  into  the  vertical  component  CI  equal  to 
OR,  and  the  horizontal  one  CK  parallel  and  equal  to  ON.  The  hori- 
zontal projection  of  CK  is  FL,  equal  to  DF,  and  compounding  it  with 
FQ,  the  resultant  FS  is  a  horizontal  line,  vertically  projected  in  CK. 

61.  The  motions  of  the  two  lines  DG,  OC,  then,  have  each  a  com- 
ponent of  vertical  translation  equal  to  OR.     The  motion  of  OC  has 
for  its  remaining  components,  DF  for  the  point  0,  and  FS  for  the 
point  C,  both  horizontal.     The  resultant  of  these  is  a  rotation  about 
the  instantaneous  axis  OP,  whose  horizontal  projection  is  T,  the  inter- 
section of  the  two  vertical  planes  normal  to  DF  and  FS.     The  re- 
maining component  of  the  motion  of  G,  is  the  horizontal  line  G  U,  and 
the  resultant  of  GU  and.  DF  is  a  rotation  of  DG  about  an  instanta- 
neous axis  determined  as  in  Fig.  10 ;  which  axis  is  also  the  vertical 
line  through  T.     And  the  rotations  of  the  two  lines  about  this  axis 
have  the  same  angular  velocity ;  for,  drawing  TS,  we  shall  have  from 
the  similar  triangles  FLS,  FDT,  in  which  DF '=  FL,  the  ratios 

TD      TD      TF 
FL  ~DF  =  FS> 

but  TFS,  TDF,  are  both  right  angles,  hence  those  triangles  are  sim- 
ilar, and  the  angles  STF,  FTD,  GTUzTQ  equal. 

Combining  this  rotation  with  the  translation,  the  final  resultant  is 
a  helical  motion  of  both  lines  and  therefore  of  the  whole  body  about 
OP. 

62.  In  the  vertical  projection,  draw  PY  parallel  to  OA,  and  PZ 
parallel  to  OC,  making  PY  equal  to  OZ,  and  the  triangles  PCY, 
AGO,  similar. 

Let  V  =  ang.  vel.  of  D        around  OA, 

V—    "      "     "  G  "       OC, 

v   =    "      "     "  D  or  G    "       OP. 

We  shall  then  have,  since  ang.  vel.  =  m*  Je  ', 

rad. 

V_OE  ON         V        OE       DT 

V  -  DG '  V  ~~  ~~  Dl  '  * "'  v   ~   ON  X  DG  ' 


But  from  aim.  triang.  ONE,  OP  A, 


RESULTANT   MOTIOH   OF   A   RIGID   BODY.  27 

and   from  sim.  triang.    TDF,  TGU. . .  —  =  — - 


PC 

OP' 


-..        .-. 

UF~  OP  X  ~AG~- 


Now  from  sim.  triang.  ACO,  PCY, 


^ 

AC~  P0 


.       -  .  -  —      - 

'*  v  ~  PC  X  OP~  OP  ~~  OP 


V        OY 
And  similarly,  —  = 


Draw  A  X,  CW9  respectively  perpendicular  to  00  and  OA  ;  then 
by  reasoning  similar  to  that  used  in  connection  with  Fig.  22,  it  may 
be  shown  that 

V  _  CP  V'_  AP 

"a      ~ 


63.  From  which  it  appears,  that  in  the  parallelogram  ZY,  of  which 
the  adjacent  sides  OZ,  OY,  are  parallel  to  the  axes  and  proportional 
to  the  angular  velocities  of  the  revolution  and  the  rotation  about  those 
axes  respectively,  the  diagonal  OP  is  parallel  to  the  instantaneous  axis 
and  proportional  to  the  angular  velocity  of 

the  rotation  about  it.  And  that  the  seg- 
ments AP,  PC,  into  which  A  C  perpendicu- 
lar to  OP  and  limited  by  the  prolongations 
of  OZ  and  0  Y,  is  divided  at  P,  are  propor- 
tional to  those  into  which  DG,  the  com- 
mon perpendicular  of  the  two  axes,  is  cut 
by  OP. 

In  Fig.  25,  as  in  Figs.  21  and  22,  the  ro- 
tation about  OC and  the  primary  revolution 
about  OA  are  in  the  same  direction.  Should 
they  have  contrary  directions,  the  argu- 
ment would  still  apply  without  change; 
and  the  modifications  in  the  diagram  caused 
by  that  change  are  shown  in  Fig.  26,  which 
being  lettered  throughout  to  correspond 
with  Fig.  25,  requires  no  explanation. 

64.  V.  Let  the  primary  motion  be  one  of 

translation,  in  which  the  paths  are  not  in  planes  perpendicular  to  the  line. 


MOTION   OF   A   KIGID   BODY 


SPACE. 


The  resultant  is  then  a  helical  motion  of  the  body  about  an  in- 
stantaneous axis  parallel  to  the  line. 

In  Fig.  27,  let  the  line  OC  be  vertical,  and  D  its  horizontal  pro- 
jection ;  the  motions  OE,  CH,  of  the  points  0  and  C,  being  parallel 
to  the  vertical  plane,  and  horizontally  projected  in  DF.  These 
motions  being  equal  and  parallel,  may  be  resolved  with  reference  to 
the  horizontal  plane  into  the  equal  vertical  components  OR,  01,  and 
the  equal  and  parallel  horizontal  ones  ON,  CK,  both  of  the  latter 
being  horizontally  projected  in  DF. 

Let  0  be  the  vertical  and  DG  the  horizontal  projection  of  a  line  of 
the  rigid  body,  and  let  OB,  GU,  be  the  corresponding  projections  of 
the  rotation  of  G  about  OC.     Now  G  has  also  a  motion  of  translation, 
of  which  OE  is  the  vertical  and  GL  the  horizontal  projection,  which 
l  H     may  be  resolved  into  a  vertical  component 

OR,  and  a  horizontal  one,  of  which  the 
projections  are  ON,  GL.  Compounding 
this  latter  with  GU,  the  resultant  is 
GS  =  GU  —  LG ;  the  vertical  projec- 
tion is  OM.  Hence  the  resultant  of  these 
horizontal  components  is  a  rotation  of 
OC  and  DG  about  the  instantaneous  axis 
whose  horizontal  projection  is  T,  and 
vertical  projection  OC;  with  which  is  to 
be  finally  compounded  the  vertical  mo- 
tion of  translation,  producing  the  helical 
resultant  motion  of  the  whole  body. 
FIG- 27-  If  OE  and  CH  coincide  with  OC,  that 

is  to  say,  if  the  line  primarily  move  endlong,  the  resultant  is,  obvi- 
ously, a  helical  motion  about  an  axis  coinciding  with  the  line  it- 
self. 

65.  VI.  Let  the  primary  motion  be  a  helical  one  about  an  axis  per- 
pendicular to  the  line. 

In  this  case  also  the  resultant  will  be  a  helical  motion  of  the  whole 
body  about  an  instantaneous  axis.  For  the  original  helical  motion  is 
compounded  of  a  rotation  about  the  fixed  axis  and  a  translation  par- 
allel to  it ;  this  rotation  being  compounded  with  that  about  the  trav- 
eling axis,  the  resultant  is  (57)  a  rotation  about  a  third  axis,  in  the 
plane  of  the  other  two,  and  passing  through  their  common  point. 
The  translation  may  again  be  resolved  into  components  respectively 
perpendicular  and  parallel  to  the  third  axis;  compounding  the  former 
with  the  rotation  atiput  that  axis,  the  resultant  is  (56)  a  rotation  about 
a  fourth  axis  parallef  to  the  third,  which  finally  compounded  with  the 


MOTION   OF   A   KIGID   BODY 


SPACE. 


component  of  translation  parallel  to  that  axis,  determines  the  instan- 
taneous helical  motion  of  the  body  about  the  fourth  axis. 

66.  Thus  in  Fig.  28,  let  the  vertical  fixed  axis  OA,  and  the  hori- 
zontal traveling  one  OC,  both  lie  in  the  vertical  plane,  and  let  OG, 
CK,  be  the  vertical   compo- 
nents  of    translation    of  the 

helical  motion  of  OC  about 
OA.  Draw  in  the  vertical 
plane  any  line  CA,  and  con- 
sider it  as  a  line  of  the  rigid 
bocly ;  its  horizontal  projec- 
tion is  AC'.  Let  C'D  be  the 
rotatory  component  of  the  mo- 
tion of  C,  in  its  helical  path, 
and  let  A'B  be  the  motion  of 
A  in  its  rotation  about  OC ': 
then  as  in  Fig.  21  we  find  OP 

the  instantaneous  axis  of  the  FIG.  28. 

resultant  rotation.  Since  P  lies  in  this  axis,  its  only  motion  is  one 
of  vertical  translation,  PY,  equal  to  OG  and  CK.  These  three  mo- 
tions may  be  resolved  into  PW,  OH,  and  CE,  perpendicular  to  OP, 
and  PZl  OX,  CF,  parallel  to  OP.  The  projections  of  06Yand  PC, 
on  a  plane  perpendicular  to  OP,  will  coincide  in  RL,  and  those  of 
the  perpendicular  components  OH,  PW,  will  coincide  in  RS.  In 
this  projection,  the  motion  of  C  will  be  represented  by  L  U,  the  re- 
sultant of  L M,  perpendicular  to  LR  and  equal  to  C  'D,  compounded 
with  LN,  the  projection  of  CE.  The  planes  normal  to  L  U  and  RS 
intersect  in  T,  which  represents  a  line  perpendicular  to  this  plane  of 
projection  and  therefore  parallel  to  OP  j  this  is  the  instantaneous  axis 
about  which  OC  and  PC  are  rotating,  and  the  helical  motion  results 
from  the  final  addition  of  the  parallel  components  OX,  CF  and  PZ. 

67.  It  will  be  seen  that  the  preceding  cases  include  every  possible 
motion  of  a  rigid  body  in  space.     For*if  the  motion  of  one  of  its  right 
lines  be  determined,  the  body  can  have  no  other  motion  except  one 
of  rotation  about  that  line  as  an  axis.     And  whatever  the  law  by 
which  the  movement  of  this  axis  may  be  governed,  its  motion  at  any 
instant  is  either  one  of  translation,  or  one  of  simple  rotation  about 
another  axis  in  the  same  or  a  different  plane,  or  a  helical  one  about  an 
axis  perpendicular  to  it.     Hence  it  appears  that  the  most  complicated 
motion  of  which  any  rigid  body  is  capable  at  any  instant,  is  a  helical 
one  about  some  right  line  :  that  is  to  say,  all  its  points  have  at  the 
instant  a   uniform   rotation   about   the  axis,  combined  with  a  uni- 


30  VAKIOUS   KINDS   OF   CONTACT  MOTIONS. 

form  motion  parallel  to   it,  their  paths  being  helices   of  the  same 
pitch. 

Sliding,  Rolling,  and  Mixed  Contact. 

68.  Sliding  Contact. — The  motions  of  a  piston  in  a  cylinder,  of  a 
journal  in  its  bearing,  and  of  a  screw  in  a  nut,  afford  instances  in 
which  the  contact  during  the  action  is  purely  sliding. 

Kegarding  closely  the  action  of  a  single  point  in  the  surface  of  the 
piston,  the  journal,  or  the  screw,  we  see  that  it  moves  over  a  certain 
path  in  the  surface  of  the  cylinder,  the  bearing,  or  the  nut,  and  that 
it  comes  into  contact,  one  after  another,  with  all  the  points  in  th^t 
path. 

In  these  instances  the  contact  of  the  two  pieces  is  superficial,  the 
moving  surface  being  identical  with  the  fixed  one.  But  one  piece 
may  touch  another  at  discontinuous  points,  or  even  in  only  one  point, 
and  yet  the  same  peculiarity  may  characterize  the  action. 

We  may  therefore  define  the  condition  of  Pure  Sliding  Contact  to  be, 
Such  relative  motion  of  two  pieces  that  every  point  of  contact  in  the  one, 
is  brought  into  coincidence  with  all  the  successive  points  in  their  order, 
of  a  line  in  the  other. 

69.  Sliding  contact  in  the  nature  of  things  as  they  are,  is  attended 
with  friction,  because  there  is  not  any  known  substance  whose  surface 
is  absolutely  smooth.     We  might  however  admit  that  friction  would 
exist  even  if  all  surfaces  were  perfectly  polished  and  free  from  asperi- 
ties.    But  its  existence  even  then  would  imply  relative  motion  of  two 
pieces  in  contact,  of  such  nature  that  a  given  point  of  one  should,  be- 
fore quitting  contact,  be  brought  into  coincidence  with  more  than 
one  point  of  the  other. 

Could  this  be  entirely  avoided,  and  the  action  be  made  such  that 
no  point  of  either  should  touch  two  consecutive  points  of  the  other, 
there  would  be  no  friction,  which  cannot  be  conceived  of  as  being  pro- 
duced by  contact  alone. 

Practically,  this  is  impossible*,  owing  to  imperfections  of  materials 
and  workmanship  :  but  the  abstract  condition,  compliance  with  which, 
if  it  were  attainable,  would  effect  the  result,  is  an  exceedingly  simple 
one. 

70.  In  Fig.  29,  let  the  tangent 
AB  remain  fixed  ;  if  the  circle  now 
rotate  about  the  fixed  centre  C,  the 
points  c,  d,  e.,  etc.,  of  its  circumfer- 
ence will  successively  be  brought 
into  coincidence  with  the  point  A 


KOLLING  CONTACT.  31 

of  the  tangent.  Or  if  the  circle  be  translated  without  rotating,  as 
shown  by  the  horizontal  arrow,  the  point  A  of  the  circumference 
will  come  into  coincidence  successively  with  the  points  c',  d',  e', 
etc.,  of  the  tangent :  in  either  case  the  action  is  one  of  pure  sliding 
contact. 

But  if  we  suppose  the  circle  to  roll,  like  a  hoop,  along  the  tangent, 
from  C  to  D,  it  is  readily  seen  that  the  point  A  will  instantly  quit 
contact,  and  the  points  c,  d,  e  of  the  circumference  will  come  succes- 
sively into  contact  with  the  tangent  at  c',  d',  e'.  As  it  is  commonly 
expressed,  the  circle  measures  itself  off  upon  the  tangent ;  thus  if  AK 
be  equal  to  the  arc  A  G,  then  the  point  G  will  come  into  contact  at 
K,  and  if  AB  be  equal  to  the  semicircumference  AGE,  then  E  will 
come  into  coincidence  with  B,  and  so  on. 

No  point  of  the  circumference  can  however  be  said  to  move  in  con- 
tact with  the  tangent.  The  circle  has  in  this  case  both  the  motions 
of  rotation  and  translation ;  and  the  movement  of  the  point  A  to  the 
left,  due  to  the  former,  is  neutralized  by  an  equal  movement  toward 
the  right,  due  to  the  latter.  That  point  is  therefore  for  the  instant 
at  rest ;  and  the  same  will  evidently  be  true  of  the  other  points  c,  d,  e, 
etc.,  as  they  successively  become  the  points  of  contact. 

71.  In  Fig.  30,  let  ML  be  the  common  tangent,  NN  the  common 
normal,  of  the  two  plane  curves  AB,  EG, 
in  contact  at  P.  Let  0,  o',  be  points  on  these 
curves  respectively,  consecutive  to  P ;  and 
let  EG  be  fixed. 

Then  if  AB  move  so  that  P  goes  in  the 
direction  PL,  the  effect  will  be  to  bring  this 
point  P  of  the  upper  curve,  into  coincidence 
with  0'  on  EG ;  but  if  P  goes  in  the  direc- 
tion PM,  the  point  o  of.  AB  will  be  brought 
into  coincidence  with  the  point  P  of  the 
fixed  curve  EG.  Now  the  motion  PL  may 
be  considered  as  either  a  translation,  or  a 

rotation  about  any  centre  D  in  JVJV.     If  it   *  [if 

be  the  former,  evidently  the  curve  AB  can-  FlG-  30- 

not  be  at  the  same  time  translated  in  the  opposite  direction.  But  in 
either  case  the  motion  of  P  can  be  neutralized  by  an  equal  and  op- 
posite motion  PM,  which  may  be  considered  as  a  rotation  about  any 
centre  (other  than  D)  in  NN,  as  for  instance  C.  The  resultant  of 
these  two  simultaneous  motions  of  AB  will  be  a  rotation  about  P  as 
an  instantaneous  centre,  the  effect  of  which  will  be  to  bring  the  point 
0  into  coincidence  with  o',  the  points  of  the  two  curves  which  now 


32  ROLLING  CONTACT. 

fall  together  at  P,  instantly  separating :  and  if  these  conditions  be 
continuously  maintained,  the  action  will  consist  of  a  rolling  contact 
between  AB  and  EG,  precisely  similar  to  that  in  the  preceding  case. 
The  point  of  contact  of  the  moving  curve  is  always  at  rest,  and  con- 
sequently the  action  is  unattended  by  friction. 

72.  It  is  apparent  that  the  same  kind  of  contact  will  ensue  if  in 
Fig.  29  the  circle  merely  rotates  about  C  as  a  fixed  centre,  while  AB 
moves  in  contact  with  it,  if  the  linear  velocities  and  directions  at  the 
point  of  tangency  are  the  same  :  so  also  in  Fig.  30,  if  the  points  of 
the  two  curves  which  fall  together  at  P,  move  in  the  same  direc- 
tion at  the  same  rate,  as  represented  by  PM,  the  consecutive  points 
o,  o'  will  come  into  coincidence,  and  the  action  will  be  that  of  roll- 
ing contact. 

It  is  also  obvious  that  these  two  figures  perfectly  represent  the  roll- 
ing of  a  cylinder  upon  a  plane,  and  of  one  cylindrical  surface  upon 
another,  the  rectilinear  elements  being  perpendicular  to  the  paper. 
These  illustrations  sufficiently  explain  the  nature  of  the  action  under 
consideration,  which  may  be  defined  as  follows  : 

Pure  Rolling  Contact  consists  in  such  a  relative  motion  of  two  lines 
or  surfaces,  that  the  consecutive  points  of  one  come  successively  into 
contact  with  those  of  the  other  in  their  order. 

73.  It  will  be  readily  seen  that  a  double  curved  surface,  like  that  of 
a  ball  or  an  egg,  can  roll  without  slipping  upon  a  surface  of  any  kind, 
plane,  single  curved,  warped,  or  double  curved,  convex  or  concave ; 
provided  that  the  curvature  of  the  rolling  surface  at  the  point  of  con- 
tact be  the  greater.     Tangency  existing  in  this  case  at  a  single  point 
only,  the  action  in  fact  consists  merely  in  the  rolling  of  a  line  of  one 
surface  in  contact  with  another  line  lying  on  the  other  surface,  and 
one  or  both  of  these  lines  may  be  of  double  curvature. 

We  are  thus  led  to  consider  this  kind  of  moving  contact  between 

lines,  in  its  most  general  aspect :  and  in 
illustration,  let  NN9  Fig.  31,  represent 
the  common  normal  plane  of  two  curves 
of  any  kind,  AB  and  EG:  ML  being 
their  common  tangent  at  P,  and  o,  o' 
points  on  those  curves,  consecutive  to  P, 
as  in  Fig.  30.  Considering  EG  as  sta- 
tionary  then,  it  appears,  from  the  forego- 
ing, that  the  rolling  of  AB  upon  it  is  effected  by  communicating  to 
that  curve  two  simultaneous  motions,  one  of  which  imparts  to  P  the 
motion  PL,  and  the  other  imparts  to  the  same  point  the  equal  and 
opposite  motion  PM.  Of  these  two  motions,  one  may,  and  the  other 


/ 

°/v 


KOLLING   CONTACT. 


33 


must,  be  regarded  as  a  rotation  about  a  line  at  a  finite  distance,  lying 
in  the  plane  NN:  their  resultant  is  a  rotation  about  an  instantaneous 
axis  also  lying  in  NN,  and  passing  through  P.  But  the  directions 
of  the  axes  of  the  component  rotations,  and  also  their  distances  from 
P,  are  arbitrary  :  hence  any  line  RPS,  drawn  through  P  in  the 
plane  NN9  may  be  assumed  as  the  instantaneous  axis ;  a  rotation 
about  which  will  bring  the  points  o,  o',  into  coincidence  as  required. 

74.  But  though  the  instantaneous  axis  must  pass  through  P,  it  is 
not  essential  that  it  lie  in  the  plane  NN.  For  if  any  line  be  drawn 
through  P,  then  both  PL  and  PM  may  be  resolved  into  components, 
perpendicular  to,  and  lying  in,  that  line.  The  latter  may  be  regarded 
as  motions  of  translation,  and,  being  equal  and  opposite,  they  will 
neutralize  each  other.  The  former  will  also  be  equal  and  opposite, 
and  regarding  one  or  both  as  rotatory,  the  irresultant  will  be  a  rota- 
tion about  the  assumed  line  through 
P,  as  an  instantaneous  axis. 

Thus  in  Fig.  32,  let  PR,  PS,  be 
the  vertical  projections  of  two  he- 
lices, lying  on  the  surfaces  of  two 
cylinders  tangent  along  PB ;  and  let 
MPL  be  the  horizontal  projection  of 
their  common  tangent,  and  NN  the 
horizontal   trace   of   their   common 
normal  plane.     It  will  be  apparent 
that  PL,  PM,  may  be  considered  as 
rotations  about  an  axis  in  NN.    But 
also,  with  reference  to  OPO,  we  may 
resolve  PL  into  the  components  PC, 
PD,  and   also  PM  into   the   com- 
ponents PE,  PG.     Then  regarding  PD,  PG,  as  motions  of  transla- 
tion, they  neutralize  each  other  :  and  PC,  PE,  may  be  regarded  as 
rotations  about  axes  lying  in  the  plane  of  which  00  is  the  trace.    These 
also  neutralize  each  other,  leaving  P  for  the  instant  at  rest  :  and  it 
will  readily  be  seen  that  as  a  matter  of  fact,  the  instantaneous  axis 
of  the  upper  cylinder,  in  rolling  around  the  lower,  must  in  the  present 
position  be  the  element  of  contact,  of  which  PB  is  the  vertical,  and 
OPO  the  horizontal  projection  ;  as  also  that  if  one  cylinder  rolls  upon 
the  other,  the  two  helices  move  in  rolling  contact. 

75.  In  so  far  then  as  the  rolling  of  any  one  line  upon  another  is  con- 
cerned, the  moving  line  may  at  any  instant  be  regarded  as  rotating 
about  any  right  line  passing  through  the  point  of  tangency.     This, 
as  above  pointed  out,  includes  the  case  of  rolling  contact  between  sur- 
3 


FIG.  32. 


34  ^  ROLLING  CONTACT. 

faces  which,  touch  each  other  in  only  one  point.  We  have  now  to 
consider  the  conditions  under  which  surfaces  having  more  than  one 
point  of  tangency  can  roll  together  without  sliding.  It  has  been 
stated  by  Prof.  Reuleaux,*  that  helicoids  and  certain  other  warped 
surfaces  are  capable  of  moving  in  perfect  rolling  contact.  From  a 
practical  point  of  view  it  is  of  comparatively  little  consequence  whether 
they  are  or  are  not  :  but  the  fact  that  such  a  statement  is  made  by 
such  an  authority,  warrants  the  presentation  of  the  following  consid- 
erations : 

76.  1.  The  rolling  of  one  surface  upon  another  is  a  motion  of  rota- 
tion about  an  instantaneous  axis,  which  must  pass  through  every 
point  of  tangency. 

2.  An  axis  is  a  right  line  ;  hence  if  there  be  more  than  one  point  of 
tangency,  all  of  them  must  lie  upon  a  rectilinear  element  of  contact. 

3.  This  rotation  about  the  instantaneous  axis  may  be  resolved  into 
two  component  motions,  of  which  one  must  be  a  rotation  about 
another  axis. 

4.  In  this  component  rotation,  each  point  in  the  moving  surface  de- 
scribes a  circle  with  plane  perpendicular  to  and  centre  in  the  second  axis; 
of  which  the  radius  is  a  perpendicular  from  the  point  to  that  axis. 

5.  The  other  component  motion  must  be  such  that  each  point  of 
tangency  shall  by  it  be  made  to  move  with  a  velocity  equal  to  that  due 
to  the  first  component  rotation,  and  in  the  opposite  direction. 

6.  If  this  second  component  be  a  motion  of  revolution,  the  axis 
must  lie  in  a  plane  normal  to  the  path  of  every  point  of  tangency,  de- 
termined as  above.     Hence  those  paths  must  be  parallel ;  which  re- 
quires the  axis  of  the  first  component  rotation  to  lie  in  the  same  plane 
with  the  common  element. 

7.  If  the  second  component  be  a  motion  of  translation,  all  the  points 
of  tangency  are  in  consequence  moving  in  parallel  directions.    There- 
fore the  axis  of  the  first  component  rotation  must  lie  in  a  plane  contain- 
ing the  common  element,  and  perpendicular  to  those  parallel  motions. 

8.  Consequently  the  axis  of  the  first  component  rotation  must  in 
all  cases  and  at  all  times  lie  in  the  same  plane  with  the  common  ele- 
ment :  and  the  motions  of  the  points  of  contact,  due  to  this  rotation, 
must  at  any  instant  be  in  parallel  directions. 

77.  9.  Now  the  effect  of  the  first  component  rotation  must  be,  to 
bring  the  consecutive  element  of  the  moving  surface  into  coincidence 
with  the  present  line  of  contact  on  the  fixed  one.  But  as  all  the  points 
of  the  line  of  contact  are  moving  in  parallel  directions,  by  virtue  of 

-"Kinematics  of  Machinery,  1876,  pp.  81,  82. 


MIXED   CONTACT.  35 

this  rotation,  the  consecutive  position  (that  is  to  say  the  consecutive 
element  of  the  moving  surface)  must  lie  in  the  same  plane  with  the 
original  one  :  thus  determining  a  plane  tangent  all  along  an  element. 

10.  Again  the  effect  of  the  second  component  must  be,  to  bring  the 
present  line  of  contact  on  the  moving  surface,  into  coincidence  with 
the  consecutive  element  of  the  fixed  one.     But  whether  this  motion 
be  one  of  translation  or  of  revolution,  it  imparts  parallel  motions  to 
all  points  of  the  common  element,  which  consequently  will  lie  in  a 
plane  containing  the  consecutive  position,  and  tangent  to  both  sur- 
faces all  along  an  element. 

11.  Pure  rolling  contact,  then,  is  not  possible  between  surfaces 
touching  each  other  in  more  than  one  point,  unless  they  arc  plane  or 
single  curved. 

12.  Nor  is  it  possible  between  all  such  surfaces.     No  argument  is 
needed  to  show  that  a  cone  cannot  roll  upon  a  cylinder,  nor  upon  an- 
other cone  unless  the  two  have  a  common  vertex.     The  rolling  of  one 
single  curved  surface  upon  another  involves  the  simultaneous  rolling 
of  both  upon  the  common  tangent  plane.     Hence  it  is  an  essential 
condition,  that  through  any  point  of  the  common  element,  it  shall 
be  possible  to  draw  two  lines,  one  upon  each  surface,  of  which  the 
developments  upon  the  common  tangent  plane  shall  coincide. 

78.  Mixed  Contact. — By  this  is  meant  a  combination  of  rolling  and 
sliding  contact ;  the  nature  of  such  action  will  be  readily  understood 
by  reference  to  Fig.  20,  if  we  suppose  the  circle  to  rotate  about  the 
centre,  and  at  the  same  time  to  be  translated,  but  in  such  manner 
that  the  points  c,  d,  e,  do  not  come  into  coincidence  with  the  points 
c'  d'  e'.  If,  for  instance,  the  circle  rotate  through  an  arc  A  G,  and  in 
the  same  time  be  translated  through  a  space  AB  or  Ad',  greater  or 
less  than  that  arc,  it  is  clear  that  though  the  action  partakes  to  some 
extent  of  the  nature  of  rolling,  yet  there  must  be  a  sliding  between 
the  circumference  and  the  tangent,  to  an  amount  depending  upon 
the  difference  between  the  arc  AG,  and  the  distance  AB  or  Ad'. 
The  resultant  of  the  two  motions  imparted  to  the  circle  will  be  a  ro- 
tation about  an  instantaneous  axis,  which  will  however  not  pass 
through  the  point  of  tangency  A  :  and  this  affords  a  means  of  ascer- 
taining whether  in  any  given  case  the  contact  motion  is  of  this  de- 
scription or  not. 

But  we  may  for  all  the  practical  purposes  of  mechanism,  disregard 
this  particular  combination  ;  and  we  shall,  in  what  follows,  take  note 
only  of  the  distinction  between  those  contact  motions  in  which  the 
action  is  purely  rolling,  and  those  in  which  it  is  not ;  all  the  latter 
being  called  cases  of  sliding  contact. 


CHAPTER    III. 


ELEMENTARY  COMBINATIONS — DRIVER  AND  FOLLOWER  ROTATING 
ABOUT  FIXED  PARALLEL  AXES — DETERMINATION  OF  VELOCITY 
RATIO  AND  DIRECTIONAL  RELATION,  1,  IN  LINK  MOTIONS  ;  2, 
IN  BAND  MOTIONS  ;  3,  IN  CONTACT  MOTIONS — CONDITIONS  OF 
ROLLING  CONTACT  AND  OF  CONSTANT  VELOCITY  RATIO — SIMI- 
LARITY OF  ACTION  IN  ALL  THE  MODES  OF  TRANSMISSION. 


Cr 


FIG.  33. 


79.  It  has  been  shown,  that  even  the  simplest  motion,  a  rectilinear 

one,  may  be  resolved  into  compo- 
nents. But  if  there  be  any  motion 
employed  in  mechanical  structures, 
entitled  to  be  called  a  simple  one  by 
reason  of  the  ease  with  which  it  can 
be  produced  and  continuously  main- 
tained, that  motion  is  certainly  the 
one  of  rotation  about  a  fixed  axis. 
Naturally  enough,  therefore,  it  is  more  frequently  met  with  than 

any  other,  and  the  very  mention  of  machinery  suggests  ideas  of  levers 

and  Avheel-work.     Nothing  can  be 

simpler  than  the   means   by  which 

rotation  is  determined,  nor  can  the 

axes  have  a  simpler  relation   than 

that  of  parallelism  :   and,   accord- 
ingly, by  far  the  largest  and  most 

important  class  of  elementary  com- 
binations   is    that   in  which    both 

driver   and  follower    rotate    about 

fixed   axes   which    are    parallel   to 

each  other.    In  the  analysis  of  their 

action  it  will  be  most  convenient  to 

consider  first  those  in  which  the 

motion    is    transmitted    by   means    of    a    link. 


FIG.  34. 


TELOCITY   RATIO 


LINK   MOTIONS. 


37 


Velocity  Ratio  in  Link  Motions. 

80.  In  Figs.  33  and  34,  let  C  and  D  be  fixed  axes,  about  which  turn 
the  levers  A  C,  BD,  whose  free  ends  are  connected  by  the  link  A B ; 
the  axes  being  perpendicular,  and  the  planes  of  motion  parallel,  to 
the  paper. 

The  motions  of  A  and  B  are  necessarily  perpendicular  to  A  C  and 
BD  respectively ;  whence  the  instantaneous  axis,  E,  is  found  by  pro- 
longing those  radii  till  they  intersect. 

From  C,  D,  E,  let  fall  the  perpendiculars  CG,  DH,  EF,  on  the 
line  of  the  link  AB  or  its  prolongation.    Also  draw  the  line  of  centres 
CD,  cutting  the  line  of  the  link,  prolonged  if  necessary,  in  the  point  /. 
Now  let 

v  —  ang.  vel.  of  A          around  C, 
v'  =  "      "     "  B  "       D, 

w  =  "      "    "  A  or  B     "       E. 


Then  by  similar  triangles, 

v^_AE_  EF 
w~~~AC~~  ~CG9 


~CG  ~~  ci 


^ 
v'~  BE~~  EF 

81.  This  result  may  be  deduced 
otherwise,  thus  :  In  Fig.  35,  let  the 
linear  velocity  of  A  be  represented 
by  AL,  perpendicular  to  AC:  then 
AM  is  its  component  in  the  line  of 
the  link,  to  which  BN  must  be 
equal.  The  direction  of  BO,  the 
motion  of  B,  is  perpendicular  to  BD, 
and  its  length  is  ascertained  by  draw- 
ing NO  perpendicular  to  AB. 

Then  by  similar  triangles  we  have 


AL  _AM 
AG~  CG' 


V   = 


__ 

=  BD  ~  DH' 


v__DH_DI 
v'-~CG~  CI 


We  have,  then,  two  simple  and  convenient  values  for  the  velocity 
ratio,  which  may  be  thus  expressed  : 


38 


DIRECTIONAL  RELATION. 


1.  The  angular  velocities  of  the  arms  are  to  each  other  inversely  as  the 
perpendiculars  from  their  centres  of  motion  upon  the  line  of  the  link. 

2.  The  angular  velocities  of  the  arms  are  to  each  other  inversely  as 
the  segments  into  which  the  line  of  the  link  cuts  the  line  of  centres. 

82.  Directional  Relation. — The  perpendiculars  CG  and  DH,  from 
the  centres  of  motion  to  the  line  of  the  link,  are  called  the  Effective 
Lever  Arms.  When  they  lie  on  the  same  side  of  AB,  as  in  Fig.  33, 
the  arms  will  turn  in  the  same  direction.  When  they  lie  on  opposite 
sides  of  the  line  of  the  link,  as  in  Figs.  34  and  35,  the  rotations  will 
be  in  opposite  directions. 

Dead  Points. — It  is  evident  that  in  the  combination  shown  in  Fig. 
33,  the  shorter  lever  BD  is  capable  of  turning  completely  round. 
When  a  lever  thus  makes  entire  revolutions,  as  is  very  often  the  case, 
it  is  usually  called  a  Crank  ;  but  whether  it  does  or  not,  it  is  obviously 
possible  for  the  system  to  come  into  either  of  the  positions  shown  in 
Figs.  36  and  37,  in  which  AB  and  BD  coincide. 


Fia.  36. 


Fio.  37. 


When  this  happens  the  system  is  said  to  be  at  a  dead  point,  because, 
the  effective  lever  arm  DH  having  disappeared,  and  the  motion  of  B 
having  no  component  in  the  line  of  the  link,  the  arm  A  C  is  moment- 
arily at  rest.  That  arm  has  a  reciprocating  circular  motion  ;  and 
the  dead  points  obviously  occur  at  the  extreme  limits  of  its  travel. 
By  way  of  distinction,  the  crank  BD  is  said  to  be  at  an  outward  dead 
point  in  Fig.  36,  and  at  an  inward  dead  point  in  Fig.  37. 

83.  Momentary  Constancy  of  Ve- 
locity Ratio. — In  general,  the  veloc- 
ity ratio  in  such  combinations  is 
varying,  as  the  ratio  between  the 
effective  lever  arms  changes  from 
instant  to  instant :  though  if,  as  in 
Fig.  13,  the  levers  are  equal  and 
parallel,  the  velocity  ratio  is  con- 
stant throughout  the  action.  But 
it  may  be  momentarily  constant 
under  other  circumstances.  ^  Thus  in  Fig.  38,  the  arms  A  C,  BD,  are 


FIG.  38. 


BAND   MOTIONS. 


39 


parallel,  but  not  equal.  The  motions  of  the  points  A  and  B  are  parallel 
to  each  other,  and  since  they  are  not  perpendicular  to  the  line  of  the 
link,  their  linear  velocities  must  be  equal,  whence  we  have  directly 


^ 

AC~  CG 


Now,  the  positions  of  the  link  consecutive  to  the  present  one  will 
be  parallel  to  AB,  and  cut  the  line  of  centres  in  points  consecutive 


FIG.  39.  FIG.  40. 

to  and  practically  coincident  with  /.     Hence  during  such  elementary 
motion  the  variation  in  the  velocity  ratio  will  be  inappreciable. 

84.  Again,  the  line  of  the  link  in  Figs.  33  and  34  is  tangent  at  F  to 
the  circle  described  by  that  point  about  the  instantaneous  axis  E. 
The  consecutive  positions  of  AB  therefore  in- 
tersect each  other  in  F :  as  may  be  seen  very 

clearly  in  Figs.  36  and  37,  where  F  coincides 
with  A,  which  is  momentarily  at  rest,  while  Hr 
BD  moves  in  either  direction  to  its  consecu- 
tive position. 

Now  in  Figs.  39  and  40,  the  foot  F  of  the 
perpendicular  upon  AB  from  the  instanta- 
neous axis,  coincides  with  /,  the  intersection 
of  the  link  with  CD  the  line  of  centres.  In 
its  consecutive  positions,  therefore,  the  line  of 
the  link  will  cut  the  line  of  centres  into  prac- 
tically the  same  segments  as  in  its  present 
one,  and  the  velocity  ratio  is  momentarily 
constant  in  this  case  also. 

Band  Motions. 

85.  Let  the  two  curved  pieces  whose  fixed 

centres  of  motion  arc  C  and  Z>,  Fig.  41,  be  FIG.  41. 

connected  by  a  flexible  and  mextensible  band  or  cord,  wrapped  upon 


40 


CONTACT   MOTIONS. 


their  convex  edges,  and  fastened  to  them  at  the  points  K  and  F.  Let 
the  lower  curve  turn  to  the  right,  as  shown  by  the  arrow  :  the  band 
will  then  be  wound  upon  AE,  and  unwound  from  BF,  thus  com- 
pelling the  upper  curve  to  rotate.  It  is  evident  that  the  parts  which 
are  wrapped  upon  the  curves  being  always  idle,  the  effective  length 
of  the  band  is  the  distance  between  the  points  of  tangency  ;  which 
may  or  may  not  change  during  the  action. 

In  the  position  shown,  the  points  of  tangency  are  A  and  B,  whose 
motions  at  the  instant  must  be  perpendicular  to  the  contact  radii,  A  C9 
BD.  Their  linear  velocities,  AL,  BO,  must  be  such  that  the  com- 
ponents AM,  BN,  along  the  line  AB,  shall  be  equal,  because  the 
band  is  inextensible  ;  the  components  AR,  BS,  perpendicular  to  AB, 
are  ineffective,  because  the  band  is  flexible.  Now  drawing  the  per- 
pendiculars CG,  DH,  from  the  centres  of  motion  to  the  line  of  the 
band,  and  also  the  line  of  centres  CD  cutting  A  B  in  /,  it  is  evident 
on  comparison  that  this  combination  at  the  instant  differs  from  that 
of  Fig.  35  in  no  particular  except  the  flexibility  of  the  connector  AB. 
And  we  shall  have,  precisely  as  in  that  case, 


CG  ~  CI 


And  also  as  in  that  case,  the  rotations  will  be  in  the  same  direction 

if  CG  and  DHlie  on  the  same  side 
of  AB,  and  in  contrary  directions  if 
they  lie  on  opposite  sides  of  that  line. 

Contact  Motions. 

86.  In  Fig.  42,  let  C  and  D  be  the  . 
fixed  centres  of  motion  of  the  two 
curved  pieces  in  contact  at  P,  their 
common  tangent  being  TT,  and 
their  common  normal  NN  :  it  is 
evident  that  if  the  upper  curve  turn 
as  shown  by  the  arrow,  the  lower 
one  will  be  compelled  to  rotate. 
Draw  the  radii  of  contact,  P  C  and 
PD  ;  then  the  motion  of  P,  con- 
sidered as  a  point  of  the  upper  piece, 
will  be  in  the  direction  PA,  perpen- 
dicular  to  PC,  while  the  direction  of 
the  motion  of  P  on  the  lower  curve  must  be  PB,  perpendicular  to  PD. 


^— 

CONTACT  MOTIONS.   mi  " 


Kesolve  P^4  into  the  components  PM,  PO;  of  these  the  former 
obviously  does  not  tend  to  move  the  lower  curve,  and  it  is  also  evident 
that  the  magnitude  of  PB  must  be  such  that  its  normal  component 
shall  be  PO;  for  were  it  greater  than  PO,  the  lower"  curve  would 
quit  contact  with  the  upper,  and  if  it  were  less  the  two  curves  would 
intersect,  both  of  which  are  contrary  to  the  hypothesis.  Draw  CG, 
DH,  perpendicular  to  NN;  then  the  triangles  PCG,  APO,  are  simi- 
lar, as  also  are  PDH,  BPO.  Draw  CD  cutting  JOT  at  /;  then  ICG, 
IDH  are  similar ;  and  letting 

v  —  ang.  vel.  around  C ; 
v'  —  "      "        "      D. 
we  have 


V'=PJB_PO 

PI)  ~DH 

Which  may  be  thus  expressed  : 

1.  The  angular  velocities  of  two  pieces  in  contact  are  to  each  other 
inversely  as  the  perpendiculars  let  fall  upon  the  common  normal  from 
the  centres  of  motion. 

2.  These  angular  velocities  are  to  each  other  inversely  as  the  seg- 
ments into  which  the  common  normal  cuts  the  line  of  centres. 

87.  Condition  of  Constant  Velocity  Ratio. — This  is  at  once  deduced 
from  the  second  of  the  above  values,  viz  ; 

v^_DI 
v'~~  Cl'' 

for  DI  +  CI  =  CD,  which  is  constant,  therefore  /  must  be  fixed,  and 
the  two  segments  at  all  times  the  same. 

In  other  words,  the  curves,  in  order  to  maintain  a  constant  velocity 
ratio,  must  be  such  that  their  common  normal  shall  always  cut  the 
line  of  centres  in  the  same  point. 

88.  Condition  of  Compulsory  Rotation. — A  rigid  link  can  both  push 
and  pull,  so  that  the  direction  of  the  driver's  motion  is  arbitrary.     A 
band  is  capable  of  pulling  only :  while  on  the  other  hand  one  piece 
can  drive  another  in  direct  contact  with  it  only  by  pushing.     In  Fig. 
43,  the  lower  piece  is  the  driver,  turning  as  shown  by  the  arrow.    Let 
x,  z,  be  points  consecutive  to  and  on  opposite  sides  of  P,  the  former 
being  in  advance.     Then  CP  is  greater  than  Cx,  but  less  than  Cz ; 
and  it  is  clear  that  in  order  to  produce  compulsory  rotation,  the 


SLIDING   COMPOHEHTS. 


curves  and  their  motions  must  be  such  that  the  contact  radius  of  the 

driver  shall  he  on  the  increase. 

89.  Rate  of  Sliding.— The  tangential  components  PM,  PL,  repre- 
sent the  linear  velocities  with 
which  the  curves  are  at  the  in- 
stant sliding  on  the  common  tan- 
gent. In  Figs.  42  and  43,  these 
components  lie  in  the  same  di- 


O  N. 


FIG.  43. 


FIG.  44. 


rection  from  P,  and  consequently  their  difference  LM  represents  the 
rate  at  which  the  two  curves  are  sliding  upon  each  other.  In  Figs. 
44  and  45  these  components  fall  in  opposite  directions,  and  the  rate 
of  sliding  is  represented  by  their  sum. 

Directional  Relation. — From  these  various  diagrams,  which  are 
lettered  to  correspond  throughout,  we  also  perceive  that  the  direc- 
tional relation  depends  upon  the  positions  of  the  perpendiculars  CG, 
DH,  with  reference  to  the  common  normal ;  if  they  lie  upon  the  same 
side  of  NN,  the  rotations  will  be  in  the  same  direction  ;  if  on  oppo- 
site sides,  the  rotations  will  be  in  contrary  directions. 

90.  Condition  of  Rolling  Contact. — In  order  that  there  may  be  no 


FIG.  46. 


sliding  between  the  two  moving  pieces,  the  tangential  components 
PM,  PL,  must  have  the  same  magnitude  and  the  same  direction,  as 
in  Fig.  46.  The  normal  components  PO  being  the  same  in  both 


CONDITION   OF   ROLLING   CONTACT.  43 

motions,  the  resultants  PA  and  PB  must  also  be  alike  in  magnitude 
and  direction  ;  and  these  being  perpendicular  to  the  contact  radii  PC, 
PD,  the  latter  must  also  coincide  in  one  right  line,  which  can  be  no 
other  than  CD. 

That  is  to  say  :  the  condition  of  rolling  contact  is,  that  the  point  of 
contact  shall  lie  upon  the  line  of  centres. 

The  curves  may  be  of  such  a  nature  that  this  condition  is  continu- 
ously satisfied,  the  point  of  contact  traveling  along  the  line  of  centres, 
and  the  velocity  ratio  varying  accordingly. 

On  the  other  hand,  that  point  may  travel  across  the  line  of  centres, 
the  action  taking  place  partly  on  one  side  and  partly  on  the  other ;  in 
which  case  the  velocity  ratio  may  or  may  not  vary ;  but  whether  it 
does  or  not,  there  will  be  more  or  less  of  sliding  between  the  curves, 
except  at  the  instant  when  the  point  of  contact  crosses  that  line. 

91.  Similarity  of  Action  in  all  the  Modes  of  Transmission, — It  will  be 
observed  that  the  common  normal  in  contact  motions  has  a  very 
striking  resemblance  to  the  lines  of  the  link  and  the  band,  previously 
discussed.     In  fact,  we  may  select  any  two  points  in  this  normal, 
and,  drawing  right  lines  from  them  to  the  centres  of  motion,  form 
thus  a  pair  of  arms  and  a  link,  which  combination  will  at  the  instant 
have  precisely  the  same  action  as  the  original  contact  pieces. 

AVo  perceive,  in  short,  that  the  motion  of  the  driver  is  transmitted 
to  the  follower  in  a  right  line,  whether  they  be  in  contact  or  not ;  and 
that  the  action  of  all  elementary  combinations  in  which  the  two  pieces 
rotate  about  fixed  parallel  axes,  is  governed  by  these  laws  : 

I.  The  angular  velocities  are  to  each  other  inversely  as  the  perpen- 
diculars from  the  centres  of  motion  upon  the  Line  of  Action ;  or,  in- 
versely as  the  segments  into  which  the  line  of  action  divides  the  line 
of  centres. 

II.  The  rotations  have  the  same  direction  if  the  centres  of  motion 
lie  on  the  same  side  of  the  line  of  action  ;  and  contrary  directions  if 
they  lie  on  opposite  sides. 

III.  The  rate  of  sliding,  in  contact  motions,  is  the  difference  of  those 
components  of  the  actual  motions  of  the  common  point  about  the  two 
centres  respectively,  which  are  perpendicular  to  the  line  of  action, 
when  they  lie  on  the  same  side  of  that  line ;  and  their  sum  if  they  lie 
on  opposite  sides. 

92.  We  may  now  proceed  either  to  examine  separately  the  action  of 
given  combinations,  or  to  ascertain,  by  the  aid  of  the  principles  thus 
far  explained,  the  forms  and  proportions  which  may  or  must  be  given 
to  the  elements  in  order  to  satisfy  assigned  conditions.     The  latter  is 
plainly  the  more  systematic  and  fruitful  course ;  and  wheels  whose 


44  COMBINATIONS  TO   BE   DISCUSSED. 

axes  are  parallel,  enter  so  largely  into  the  composition  of  machinery 
of  all  descriptions,  that  the  fact  is  a  sufficient  reason  for  at  present 
confining  our  attention  to  the  transmission  of  motion  by  direct  con- 
tact ;  considering  first  those  cases  in  which  the  action  is  purely  roll- 
ing, and  taking  up  afterward  those  in  which  the  contact  is  of  the 
mixed  variety. 


CHAPTER  IV. 


ROTATION    BY    ROLLING    CONTACT,     FIXED    PARALLEL    AXES. 


1.  VELOCITY  RATIO  CONSTANT. — Toothless  Circular  Wheels.     Friction  Gearing. 

2.  VELOCITY    RATIO    VARYING. — The  Logarithmic  Spiral,  and  Toothless  Lobed 

Wheels  formed  from  it.  The  Rolling  Ellipses,  and  Lobed  Wheels  derived 
from  them  by  Contraction  of  Angles.  Interchangeable  Multilobes,  derived 
from  Ellipse  and  from  Logarithmic  Spiral.  Irregular  Lobed  Wheels. 

1.    Velocity  Ratio    Constant. 

93.  If  the  teeth  of  a  pair  of  ordinary  circular  spur  wheels  be  in- 
definitely reduced  in  size  and  in- 
creased in  number,  they  will  ulti- 
mately become  mere  lines,  or  rec- 
tilinear elements    of    cylinders  of 

revolution,  as  shown  in  Fig.  47. 

These  are  technically  called  the 
Pitch  Surfaces  of  the  wheels.  Evi- 
dently they  are  capable  of  moving 
in  perfect  rolling  contact  about 
their  axes,  the  linear  velocities  of  FlG-  47- 

their  circumferences  being  the  same,  and  the  angular  velocities  in- 
versely proportional  to  the  radii,  or  to  the  original  numbers  of  the 
teeth.  Which  agrees  with  the  deductions  of  the  preceding  chapter  ; 
the  condition  of  rolling  contact  is  (90),  that  the  point  of  tangency 
shall  always  be  upon  the  line  of  centres  :  and  the  condition  of  a  con- 
stant velocity  ratio  is,  (87),  that  the  common  normal,  which  passes 
through  that  paint,  shall  always  divide  the  line  of  centres  into  the 
same  segments.  Hence  in  order  to  satisfy  both  conditions  at  once, 
the  contact  radii  must  be  constant ;  or  in  other  words  the  contact 
curves  must  be  circles  whose  centres  are  in  the  axes. 

94.  But  it  is  important  to  note  that  although  circles  are  the  only 
curves  which  can  move  in  rolling  contact  about  fixed  axes,  yet  one  has 


46  FRICTION   GEARING. 

no  tendency  to  drive  the  other.  The  condition  of  compulsory  rota- 
tion is,  (88),  that  the  contact  radius  of  the  driver  shall  be  on  the  in- 
crease ;  which  is  here  impossible. 

And  again,  in  Fig.  47,  let  the  upper  circle  turn  as  shown  by  the  ar- 
row, the  motion  of  the  contact  point  being  represented  by  PB  ;  then 
this  motion,  being  tangent  to  both  circles,  has  no  normal  component 
at  all.  Consequently  the  driver  tends  merely  to  slip  upon  the  fol- 
lower, not  to  cause  the  latter  to  move.  And  it  is  a  well-known  fact, 
that  two  polished  rollers  cannot  be  used  to  transmit  rotation  with  a 
constant  velocity  ratio  :  the  more  nearly  they  approach  theoretical 
perfection  as  cylinders,  the  farther  they  depart  from  practical  perfec- 
tion as  wheels,  and  if  motion  be  transmitted  at  all,  it  is  by  reason  of 
adhesion  between  them. 

95.  Friction  Gearing. — Nevertheless  such  toothless  wheels  arc  quite 
extensively  used  in  practice,  constituting  what  is  known  as  Friction 
Gearing.     The  axes  arc  pressed  together  with  considerable  force,  and 
in  order  to  secure  the  greater  adhesion,  the  peripheries  in  contact  are 
usually  made  of  different  materials  :    thus  if  one  wheel  be  of  iron, 
the  other  may  be  made  of  wood,  or  have  its  rim  covered  with  leather 
or  india-rubber.      There  is,  of  course,  always  a  possibility  that  such 
wheels  may  slip  upon  each  other  ;  in  point  of  fact  they  usually  do, 
and  the  amount  of  slip,  moreover,  varies  under  different  conditions  of 
pressure  and  speed.     There  is,  then,  no  certainty  that  a  constant  ve- 
locity ratio  will  be  preserved,  and  they  cannot  be  employed  where 
that  is  imperative  ;  but  there  arc  many  cases  in  which  it  is  not. 

96.  Grooved  Friction  Gearing. — Another  arrangement  of  friction 

gearing,  very  generally  used  in  hoisting 
machines,  especially  for  mining  pur- 
poses, is  shown  in  Fig.  48  ;  the  two 
wheels  having  a  series  of  angular 
grooves  turned  in  their  peripheries,  so 
M  that  by  pressing  them  together  an  effec- 
tive degree  of  adhesion  is  secured  :  both 
wheels  are  usually  made  of  cast  iron. 
The  action  is  evidently  attended  with 
considerable  sliding  in  any  case,  and  the 
amount  may  vary  between  wide  limits  : 
FIG.  48.  but  for  such  work  a  constant  velocity 

ratio  is  not  requisite,  and  the  possibility  of  slipping  is  a  practical  ad- 
vantage, as  it  reduces  the  shock  if  the  machine  be  suddenly  started 
or  stopped,  thus  diminishing  the  risk  of  breakage.  The  velocity  ra- 
tio when  in  full  work  is  substantially  the  same  as  that  between  two 


VELOCITY   KATIO   VARYING. 


pitch  cylinders  whose  line  of  tangency  is  midway  between  the  tops 
and  bottoms  of  the  grooves,  as  L M  in  the  figure. 

2.    Velocity  Ratio   Varying. 

97.  If  the  contact  curves  are  not  required  to  maintain  a  constant 
velocity  ratio,  then,  since  the  point  of  tangency  must  still  be  always 
upon  the  line  of  centres,  the  contact  radii  will  vary.     But  their  sum, 
evidently,  remains  constant,  if  the  point  of  contact  lies  between  the 
centres  of  motion  ;  or  if  those  centres  lie  on  the  same  side  of  that 
point,  the  difference  of  the  contact  radii  must  be  constant. 

And  by  reference  to  Fig.  46,  it  will  be  seen  that  if  PR,  PS,  be  two 
equal  arcs  of  the  curves,  the  latter  must  be  of  such  form  that 

CR  +  DS  =  CP  +  DP  =  CD. 

We  proceed  now  to  show  how  some  curves  may  be  constructed,  which 
will  satisfy  the  above  condition,  and  the  manner  in  which  various 
changes  in  the  velocity  ratio  may  be  effected  by  means  of  them. 

98.  The  Logarithmic  Spiral.— In  Fig.  49,  let  the  parallels  YU,  TW, 
be  cut  at  any  angle  by  TB,  and  , 

perpendicularly    by  the   other  I        i 

parallels  CD,  IN,  BN,  etc. 
About  C  as  a  centre,  describe 
an  arc  with  radius  CE  —  A  M  ; 
and  about  P  as  centre,  describe 
an  arc  with  radius  PA,  cutting 
the  first  in  E,  thus  constructing 
the  triangle  CPE.  With  cen- 
tres C  and  E,  and  radii  CG, 
EG,  respectively  equal  to  BN 
and  AB,  describe  arcs  intersecting  in  G.  Continuing  in  a  similar 
manner,  the  result  is  the  formation  of  the  broken  line  RP  V. 

Now  let  us  imagine  C  to  turn  first  about  P  until  P^coincides  with 
PA,  and  then  about  A  until  EC  coincides  with  AM;  then  RPV 
will  have  the  position  shown  in  dotted  lines,  the  triangle  EP C  ap- 
pearing as  A  OM.  After  which  in  a  similar  manner  G  may  be  brought 
to  coincide  with  B,  the  point  C  meantime  advancing  to  N,  and  soon. 

99.  The  polygon  7?PFthus  travels  with  a  sort  of  imperfect  rolling 
action,  measuring  off  its  perimeter  on  TB,  since  PE,  EG,  are  by  con- 
struction equal  to  PA,  AB,  respectively;  also  PV  =  PH,  etc. 

Now  if  the  subdivisions  on  TB  be  made  indefinitely  minute,  so  that 
A  and  H  are  points  consecutive  to  P,  then  RP  Fwill  become  a  curve 


FIG.  49. 


48" 


LOGARITHMIC   SPIRAL. 


to  which  TB  is  tangent  at  P.  This  curve  will  not,  however,  pass 
through  the  angles  of  the  polygon  here  shown  ;  since  the  arc  EP,  for 
instance,  must  be  equal  to  AP,  the  angle  ECP  will  in  the  curve  be 
less  than  it  now  is. 

By  reason  of  the  parallels  LH,  CD,  etc.,  the  angles  LHT,  CPH, 
MAP,  etc.,  are  equal  ;  hence  the  curve  must  be  such  that  its  tangent 
at  any  point  shall  make  a  constant  angle  with  the  radiant  from  C  to 
that  point.  It  is  apparent  from  the  mode  of  derivation  that  the  radi- 
ants from  C  will  increase  as  we  go  from  CP  to  the  left :  and  these 
conditions  determine  the  curve  to  be  the  logarithmic  or  equiangular 
spiral,  of  which  C  is  the  pole  and  TB  the  tangent  at  P. 

Supposing  this  curve  to  replace  RP  V,  the  rolling  action  will  be 
perfect,  C  traveling  in  a  right  line  to  M  and  N9  while  E  and  G  go  to 
A  and  B. 

100.  The  same  process  being  repeated  below  TB,  with  D  as  the 
pole,  we  shall  have  a  polygon  of  the  same  description  as  RPV,  but  in 
a  reversed  position.  "We  have  in  this  second  one,  PF  equal  to  PA, 
and  DF  equal  to  IA.  In  the  upper  one,  PE  =  PA,  and  CE '—  AM-, 
hence  if  the  two  rotate  in  opposite  directions  about  C  and  D  as  fixed 
centres,  E  and  F  will  come  together  at  S  on  the  line  CD.  So  in  like 
manner  will  G  and  Z,  or  F  and  H,  meet  upon  CD. 

It  is  apparent  that  these  two  broken  lines  will  at  the  limit  become 
parts  of  equal,  similar,  and  opposite  equiangular  spirals.  The  curves 
cannot  be  constructed  in  this  manner,  except  approximately  :  but  this 
mode  of  derivation  exhibits  in  the  clearest  light  the  fact  that  they  will 
move  in  perfect  rolling  contact  about  fixed  axes  passing  through  their 
poles.  Referring  the  reader  to  the  Appendix  for  detailed  instructions 
relating  to  the  accurate  delineation  of  these  and  other  curves,  we 
proceed  to  consider  the  conditions  which  may  be  assigned  in  some  ap- 
plications of  this  spiral  in  mechanism. 

101.  Since  the  curve  does  not  return  into 
itself,  continuous  rotation  cannot  be  trans- 
mitted from  one  shaft  to  another  by  means  of 
a  single  pair  of  the  spiral  arcs.  But  for 
reciprocating  circular  motions  they  are  well 
adapted  ;  of  which  two  cases  are  illustrated  in 
Figs.  50  and  51. 

In  the  former,  let  the  centres  C  and  D  be 
given  ;  and  suppose  it  to  be  required,  that  while 
the  upper  piece  rotates  through  a  given  angle, 
the  velocity  ratio  shall  vary  between  two  as- 
FIG.  50.  signed  limits. 


LOGAKITHMIC   SPIEAL   CAMS.  49 

Divide  CD  at  A  into  two  segments,  whose  ratio  is  one  limit,  and 
at  G  into  two  others  whose  ratio  equals  the  other  limit.  Set  off 
the  given  angle  A  CB,  and  make  CB  equal  to  CO.  We  have  then  two 
given  radiants  CA,  CB,  and  the  pole  C,  to  construct  the  spiral  AB. 
The  other  sector  is  a  portion  of  the  same  curve  ;  of  which  one  radiant 
is  AD,  and  the  other  ED,  is  equal  to  GD  :  whence  the  angle  EDA  is 
determined  by  simply  cutting  the  spiral  by  arcs  described  about  the 
pole  D,  with  those  given  radii. 

Letting  v  =  ang.  vel.  of  Upper  Sector, 

v'  =    "     "       Lower      " 
we  have  in  the  position  shown, 

t    =  AD  - 

v'       AC' 

and  when  J&  and  E  meet  at  G, 


_  ^ 

v1  "  ~GC  ' 

102.  It  may  be  required  that  the  two  shafts  shall  turn  through 
equal  angles.     In  that  case  one  limit 

may  be  assigned  to  the  velocity  ratio, 
but  the  other  will  of  necessity  be  the 
reciprocal  of  the  first. 

For,  as  shown  in  Fig.  51,  we  must 
have  CB  =  AD  ;  also  the  angles 
BCA,  EDG,  are  to  be  equal,  whence,  FIG.  51. 

from  the  nature  of  the  curve,  ED  =  AC,  and  the  velocity  ratio  will 
vary  between  the  limits, 

v_    =  AD     v  __  GD  __  AC  . 
v'    ~  AC9   v'    '  GC      AD' 

of  which  the  one  is  determined  when  the  other  is  assigned. 

103.  Lobed  Wheels.  —  In  the  transmission  of  continuous  rotation,  it 
may  be  required  that  a  certain  number  of  determinate  changes  in  the 
velocity  ratio  shall  take  place  during  the  revolution  of  one  of  the 
shafts.     If  the  same  number  of  changes  is  to  occur  during  the  revolu- 
tion of  each  shaft,  the  equiangular  spiral  may  be  employed  as  follows  : 

In  Fig.  52,  let  C,  D  be  the  given  centres,  and  let  the  angles  ACB, 
ADE,  each  be  equal  to  60°.     Make  DE  -  AC,  and  CB  =  AD  :  con- 
struct the  equal  spiral  arcs  AE,  AB  ;  draw  AH,  BF,  similar  curves 
4 


50 


LOBED   WHEELS. 


symmetrically  placed  with,  reference  to  AD  and  BC.     The  velocity 
ratio  in  the  position  shown  is 


AC 


when  B  and  E  meet  at  G,  the  ratio  will  be 


-'      -in  Fig.  51. 


The  angles  EDH,  A  CF,  each  including 
one-third  of  a  circumference,  the  pair 
of  trilobes  may  be  completed  as  shown. 
They  will  roll  together,  the  velocity  ratio 
varying  from  maximum  to  minimum  and 
back  again,  three  times  in  each  revolution 
of  each  shaft. 

In  this  construction  the  limits  are  neces- 
sarily reciprocals  of  each  other,  so  that  only 
one  of  them  is  arbitrarily  assignable. 

The  angles  A  CB9  ADE,  may  include  any 
aliquot  part  of  the  circle  whose  denominator 
is  even  ;  and  in  this  way  pairs  of  wheels 
may  be  made  with  any  desired  number  of 
lobes,  which  will  roll  together.     In  Fig.  53 
those  angles  are  each  equal  to  180°,  the  re- 
suit  being  a  pair  of  unilobes  ;  in  Fig.  54, 
they  are  made  equal  to  90°,  for  constructing  bilobed  wheels. 
It  is  to  be  observed  that  the  wheels  laid  out  as  above  will  work  to- 


52. 


FIG.  53. 


FIG.  54. 


gether  only  in  equal  and  similar  pairs,  trilobe  with  trilobe,  one  uni- 
lobe  with  another  one,  and  so  on. 


THE   KOLLItfG   ELLIPSES. 


51 


FIG.  55. 


104.  The  Rolling  Ellipses. — Two  equal  and  similar  ellipses,  of  what- 
ever eccentricity,  will  also  work  in  per- 
fect rolling   contact,    each    revolving 

about  one  of  its  foci  as  a  fixed  centre  ; 
the  distance  between  the  centres  being 
equal  to  the  major  axis. 

In  Fig.  55,  let  C  and  D  be  the  fixed 
centres,  A  and  B  the  free  foci,  and  E 
a  point  of  contact.  It  is  to  be  proved 
that  E  will  always  lie  upon  the  line  of 
centres  CD. 

From  E  draw  lines  to  both  foci  of 
each  ellipse.  Then  since  the  common 
tangent  FG  makes  equal  angles  with 
all  these  lines,  we  have  AEG  =  BEF, 
and  DEG  =  CEF,  therefore  AEB, 
CED,  are  both  right  lines. 

Also,  EG  +  EA  =  EC  +  ED  =  EB 
+  ED.  Consequently  AB  =  CD,  and  EG—  EB. 

The  latter  equality  shows  that  the  elliptical  arcs  EL,  EM,  meas- 
ured from  any  point  of  tangency  to  those  extremities  of  the  major 
axes  which  will  come  together,  are  equal.  The  rolling  action  is  there- 
fore perfect. 

105.  From  the  fact  that  AB  is  constant, -we  are  enabled,  when  the 
rolling  ellipses  are  fixed  on  the  ends  of  shafts  which  overhang  their 
bearings,  to  connect  the  free  foci  by  means  of  a  link,  as  shown  in  the 
figure,  which  is   sometimes  advantageous  in  practice,  as  will  subse- 
quently appear,  even  when  the  ellipses  are  provided  with  teeth.   These 
free  foci,  obviously,  describe  circles  of  which  the  radius  is  A  C. 

106.  During  each  revolution  of  either  ellipse,  the  velocity  ratio  va- 
ries once  from  maximum  to  minimum  and  back  again.     The  limits 
are,  evidently,  mutually  reciprocal,  and  determined  by  the  ratio  of 
the  segments  into  which  either  focus  cuts  the  major  axis.     Hence  if 
one  limit  be  assigned,  and  the  distance,  between  the  axes  given,  it  is 
only  necessary  to  divide  that  distance  into  two  parts  whose  ratio  is  the 
assigned  limit :  we  have  then  the  major  axis  and  the  foci,  whence  the 
ellipses  are  determined. 

107.  Transformation    of   Rolling   Curves — Contraction   of  Angles. — 
If  any  two  curves  are  capable  of   rotating  in  rolling  contact  about 
fixed  axes,  producing  a  given  variation  in  the  velocity  ratio  during 
definite  angular  movements,  they  may  be  transformed  into  two  others, 
also  rolling  in  contact  about  the  same  axes,  and  producing  the  same 


CONTKACTIOtf  OF  ANGLES. 


FIG.  56. 


FIG.  57. 


variation  in  the  velocity  ratio,  during  angular  movements  which  are 
greater  or  less  than  the  original  ones  in  any  assigned  ratio.  This  is 
effected  by  a  process  called  the  Contraction  of  Angles,  since  usually 
the  angular  movement  of  the  derived  curves  is  less  than  that  of  the 
original  ones  during  the  same  variation  in  the  velocity  ratio. 
In  Fig.  56,  let  ^4.  be  the  point  of  contact  of  the  rolling  curves  AH, 
AI,  which  turn  with  given  angular  velocities 
about  C  and  D  :  then  at  the  end  of  a  definite 
time  the  points  /  and  H  will  meet  at  G.  In 
Fig.  57,  let  the  lengths  of  the  radiants  be  the 
same  as  in  Fig.  56,  but  the  angles  included  be- 
tween them  only  half  as  great.  Let  these  new 
curves  turn  with  the  same  angular  velocity  as 
the  original  ones  :  then  in  half  the  time,  H  and  /  will  also  meet  at  G. 
All  the  angles  being  reduced  in  the  same  proportion,  while  the  radi- 
ants remain  the  same,  it  follows  that  intermediate  points,  as  E,  B, 
which  meet  on  CD  in  the  first  pair,  will  do  the  same  in  the  second  ; 
which,  therefore,  will  move  in  pure  rolling  contact. 

108.  Lobed   Wheels  Derived  from  the   Ellipse.— -By   applying  the 
above  process  to  the  rolling  ellipses,  pairs  of  wheels  may  be  con- 
structed which  will  roll  together,  the  velocity  ratio  varying  between 
the  same  limits  and  according  to  the  same  law  as  in  the  ellipses  them- 
selves, but  the  maxima  and  minima  recurring  two,  three,  or  any  de- 
sired number  of  times  in  each  revolution. 

The  operation  is  illustrated  in  Fig.  58,  in  which  C  is  a  focus  of  the 

ellipse  whose  major  axis  is  AB. 
For  convenience  of  construction, 
a  circle  is  described  around  C,  of 
which  the  upper  half  is  divided 
into  any  number  of  parts  at  /, 
77,  ///,  etc.,  and  the  lower  left- 
hand  quadrant  is  divided  in  a 
similar  manner,  at  1,  2,  3,  etc. 
Radii  are  drawn  to  the  points  of 
division,  the  upper  ones  cutting 
the  ellipse  at  D,  E,  F,  etc. :  and 
the  segments  intercepted  between 
the  focus  and  the  curve  are  set  off 
from  C  on  the  corresponding  radii  in  the  quadrant  below,  as  Cd  =  CD, 
on  (71,  Ce  —  CE,  on  02,  and  so  on,  CB  going  to  CH. 

109.  Regarding  the  ellipse  as  a  unilobe,  is  is  evident  that  the  curve 
AeH,  thus  determined,  is  one  fourth  of  the  outline  of  abilobed  wheel, 


SIMILAR   LOBED   WHEELS. 


which  when  completed  will  roll  in  contact  with 
similar  one,  as  shown  in  Fig.  59. 
By  contracting  the  angles  to  one 
third  instead  of  to  one  half,  as 
shown  in  Fig.  GO,  the  contour  of 
a  trilobe  is  determined,  the  com- 
plete wheel  with  its  rolling  mate 
being  shown  in  Fig.  61 ;  and  in 
like  manner  wheels  with  any 
number  of  lobes  may  be  con- 
structed. 

110.  But  these  wheels,  like 
those  previously  constructed  from 
the  equiangular  spiral,  will  Work 
only  in  equal  and  similar  pairs, 
bilobe  with  bilobe,  trilobe  with 
trilobe,  and  so  on  :  consequently 
they  can  only  be  used  when  the 
number  of  changes  in  the 
velocity  ratio  is  to  be  the  same 
during  each  revolution  of  each 
shaft.  Now,  the  above  pro- 
cess may  be  so  modified  that 
from  a  pair  of  rolling  ellipses 
a  dissimilar  pair  of  wheels  may 
be  derived,  as  for  instance,  a 
bilobe  rolling  with  a  trilobe,  or 


53 


FIG.  59. 


FIG.  60. 


FIG.  81. 


a  unilobe  with  a  wheel  of  two  or  more  lobes  ;  so  that  the  numbers  of 
changes  in  the  velocity  ratio  will  be  different  in  the  two  revolutions. 


54 


DISSIMILAR   MULTILOBES. 


The  eccentricity  of  the  original  ellipses,  however,  is  not  entirely  arbi- 
trary, but  will  vary  between  limits  determined  by  the  conditions  of 
each  particular  combination. 

The  modification  just  mentioned  consists  merely  in  this,  that  a  part 
only,  instead  of  the  whole,  of  the  elliptical  outline,  is  transformed  into 
a  new  curve.  It  will  be  readily  apprehended  that  equal  arcs  of  two 
rolling  ellipses  may  subtend  at  the  fixed  foci  unequal  angles,  the  ratio 
of  whose  magnitudes  shall  be  thafc  of  two  integers  :  and  that  when 
this  is  the  case,  those  arcs  may  be  transformed  as  above  into  portions 
of  the  contours  of  lobed  wheels. 

111.  For  example,  in  Fig.  62,  let  C  be  the  fixed  and  A  the  free 

focus  of  the  ellipse  whose  major 
axis  is  HP  ;  let  D  be  the  fixed  focus 
of  its  rolling  mate,  and  B  and  E, 
two  points  which  will  meet  at  G, 
the  elliptical  arcs  PB,  PE,  being 
equal.  Now  if  the  angles  BCP, 
EDP,  are  to  each  other  as  3  to  2, 
then  by  contracting  them  in  the 
same  proportion,  the  former  may 
be  reduced  to  90°  and  the  latter  to 
60°.  Thus  the  elliptical  arc  PB 

„  ^    N.  .y,  f..  will  be  transformed  into  PI,  one 

flu    ^^^^§}-^^^^  fourth  of  the  outline  of  a  bilobe, 

and  PE  into  PM,  one  sixth  of  the 
contour  of  a  trilobe,  and  these  new 
curves  will  roll  together. 

Let  0  be  the  free  focus  of  the 
second  ellipse,  then,  drawing  OE 
and  AB,  it  is  evident  that  the 
angles  EDP,  BA  C,  are  equal. 

If,  therefore,  on  the  indefinite 
right  line  HD, we  construct  any  two 
angles  BAG,  BCD,  whose  magnitudes  have  the  ratio  of  2  to  3,  and 
produce  the  lines  which  limit  them,  till  they  intersect  at  B;  then 
from  the  ellipse  whose  foci  are  A  and  C,  and  major  axis  is  equal  to 
AB  +  BC,  a  bilobe  and  a  trilobe  may  be  derived,  which  will  roll 
together. 

The  velocity  ratio  in  the  position  shown,  is 


PD 

PC 


CH 
TC 


LOBED   WHEELS   DERIVED   PROM   ELLIPSES. 

and  when  B  and  E  meet  at  G,  it  will  be 


55 


^ 
JO        BC~    BC 


112.  In  Fig.  62,  the  angles  B CP,  BA  C,  arc  both  obtuse  ;  but  the  con- 
struction holds  true  if  they  be  made  acute,  as  in  Fig.  63.    But  whereas 

they  were  contracted  in  the  first  case, 
they  have  in  the  second  case  to  be  ex- 
panded  in  order  to  produce  the  re- 
quired contours  of  the  lobed  wheels ; 


a  circumstance  which  points  directly  to  the  conclusion  that  there 
must  be  one  ellipse  so  proportioned  that  it  is  not  necessary  to  do 
either.  This  is  shown  in  Fig.  64,  where  the  angle  BCP  being  90°, 
and  EDP,  or  its  equal  BAG',  being  60°,  the  equal  elliptical  arcs  PB 
and  PE  form  at  once  portions  of  the  outlines  of  the  bilobe  and  the 
trilobe  respectively. 

From  these  same  ellipses  another  pair  of  dissimilar  multilobes  can 
be  constructed.  Erecting  the  perpendiculars  A  I,  DK,  and  drawing 
CI,  OK,  the  elliptical  arcs  PBI,  PEK,  are  equal.  But  the  angles 
ICP,  KDO,  measuring  120°  and  90°  respectively,  are  to  each  other 
in  the  ratio  of  4  to  3  :  therefore,  contracting  them  to  one  half,  as  in 


56 


GENERAL  PROCESS   OF  CONSTRUCTION. 


Fig.  65,  we  have  portions  of  the  contours  of  a  trilobe  and  a  quad- 

rilobe  which  will  work  in  rolling  contact. 

113.  In  general,  then,  we  may  pro- 
ceed as  follows  :  If  it  be  required  to 
construct  a  wheel  with  m  lobes,  to  roll 
with  another  having  n  lobes,  supposing 
m  to  be  the  greater  number  :  Assuming 
any  two  points  A,  C,  on  the  indefinite 
right  line  HD,  Fig.  66,  construct  the 
angles  BCD,  BAD,  with  magnitudes 
whose  ratio  is  that  of  m  to  n,  the  lines 
AB,  CB9  intersecting  at  B.  Then  the 
sum  of  those  two  lines  will  be  the  major 
axis,  and  the  points  A  and  C  the 
foci,  of  an  ellipse  from  which  the 
wheels  can  be  constructed  as  above  ex- 
plained. 

Other  pairs  of  angles  having  the 
same  ratio  may  be  constructed,  deter- 
mining the  points  I,  G,  etc.  :  these 
points  will  lie  in  a  curve,  AID,  and  it 
is  evident  that  if  from  any  point  of  this 
curve,  lines  be  drawn  to  A  and  C,  they 

will  form  with  HD,  angles  in  the  required  ratio,  and  their  sum  may 

be  taken  as  the  major  axis  of  the  proposecT  ellipse  :  as  for  instance, 

UP  =  EA  +  EC,  if  E  be  the  point  selected. 
114.  The  curve  itself  may  be  constructed  either  by  determining  a 

sufficient  number  of  points 

as  above  described,  or  by 

aid  of  the  following  prop- 
erty.    The  centre  of  the 

circle  circumscribing  any 

one  of  the  triangles  whose 

base  is  AC,  and  vertex  in 

the  curve,  as  for  example  H 

ABC,   must    lie   on  the 

line  LM  perpendicular  to  and  bisecting  A  C. 
Now  let  the  angle  BCD  =  m, 
then     "       "      BAG  =11, 
and      "      "      ABC=m-n-, 

and  in  the  circumscribing  circle 

we  shall  have 


FIG.  66. 


LIMIT   OF  ECCENTRICITY.  .  57 

m  —  n  measured  by  -J  arc  A  Cy 
n          "  "  4-   "    CB, 

m          "  "I    "    ACB: 

that  is  to  say, 

arc  CB  :  arc  A  C  :  :  n  :  m  —  n. 

If  then  we  had  taken  the  centre  0  at  .pleasure  on  LM,  and,  de- 
scribing the  circle  through  A  and  G,  had  set  off  the  arc  CB  in  the 
above  ratio  to  the  arc  A  C,  the  point  B  would  have  been  thus  de- 
termined. 

115.  Now,  this  curve  springs  from  the  point  A  ;  for  when  m 
becomes  180°,  as  shown  in  the  two  small  diagrams  marked  x  and  z, 
A  C  will  divide  the  circle  whose  centre  is  0,  into  segments  such  that 
CRA9  above  A  C,  is  to  ASC  below  that  line,  as  n  is  to  m  —  n  :  and  the 
curve  will  be  tangent  at  A  to  that  circle. 

The  limit  D  of  the  curve  will  be  reached  when  m  becomes  zero  : 
that  is  to  say  when  the  radius  of  the  circle  A  CB  becomes  infinite,  in 
which  case  the  circle  itself  will  coincide  with  the  right  line  A  CD. 
We  shall  then  have 


But 

AD  +  CD  =  2  CD  +  AC 


m  —  n 


=     AC 

\m  —  n 

or, 


+  AC 


m  —  n 
whence 

AC  m  —  n 

AD  +    CD  ~~  m  +  n  * 

We  have,  therefore,  this  limit  to  the  eccentricity  of  the  ellipses 
from  which  dissimilar  wheels  with  assigned  numbers  of  lobes  can  be 
derived,  viz  :  that  the  distance  between  the  foci,  divided  by  the 
major  axis,  must  be  greater  than  the  difference  of  the  numbers  divided 
by  their  sum. 

116.  Each  wheel  derived  from  the  ellipse  has  thus  far  been  sup- 
posed to  consist  of  two  or  more  lobes.  But  the  construction  last  dis- 


58 


UNILOBES  WITH   MULTILOBES. 


cussed  will  hold  good  when  n  is  to  m  in  the  ratio  of  unity  to  any 
integer ;  therefore  by  expanding  the  elliptical  arc  subtending  the 
angle  n,  until  the  latter  equals  180°,  it  will  be  seen  that  a  unilobe 
may  be  formed,  which  will  roll  with  a  wheel  having  any  given  num- 
ber of  lobes.  This  is  sufficiently  illustrated  by  Figs.  67  and  68, 
which  represent  respectively  a  unilobe  rolling  with  a  bilobe  and  one 
rolling  with  a  trilobe. 

In  the  former  case,  it  will  be  observed  that  since  the  angles  ABC, 


BAC,  are  each  equal  to  £  BCP,  the  curve,  AID  of  Fig.  66,  in  Fig. 
67  becomes  the  semicircle  ABG,  of  which  the  centre  is  C  and  radius 
AC,  the  distance  between  the  foci  of  the  primitive  ellipse.  The 
elliptical  arc  PB,  becomes  by  expansion  the  curve  PLA,  forming  half 
the  outline  of  the  unilobe  ;  while  the  equal  arc  PE  is  expanded  into 
PM,  bounding  one  quadrant  of  the  bilobe. 

117.  Interchangeable  Multilobes,— It  appears,  then,  that  from  a 
given  ellipse  there  can  be  constructed  pairs  of  similar  wheels  having 
any  number  of  lobes  ;  and  if  the  eccentricity  be  great  enough,  several 
different  pairs  of  dissimilar  wheels  may  be  derived  from  the  same 
ellipse.  But  these  wheels  will  work  only  in  pairs  as  constructed,  and 
are  not  interchangeable ;  thus,  the  bilobe  which  rolls  with  a  trilobe, 


HOLDITCH'S  LNTEECHANGEABLE  MULTILOBES. 


59 


is  not  the  same  as  the  one  which  works  with  a  unilobe,  though  derived 
from  the  same  ellipse,  and  if  a  unilobe  and  a  trilobe  be  derived  from 
the  same  primitive  ellipse,  they  will  be  different  from  either  of  the 
others. 

There  are,  however,  two  methods  by  which  a  series  of  lobed  wheels 
may  be  constructed,  which  are  interchangeable,  any  one  working  in 
perfect  rolling  contact  with  any  other  one  of  the  series. 

118.  In  the  first  of  these  methods,  which  was  discovered  by  the 
Rev.  ITamnett  Holditch,  the  multilpbes  are  derived,  each  from  one  of 
a  series  of  different  ellipses.     Their  rolling  properties  do  not  appear 
to  admit  of  geometrical  demonstration,  but  the  process  of  construct- 
ing them  is  simple  and  practical. 

A  series  of  ellipses  is  first  made,  having  the  same  foci,  but  with 
minor  axes  which  are  to  each  other  as  1,  2,  3,  4,  etc.  Then  the 
primitive  ellipse,  whose  minor  axis  is  1,  will  roll  with  a  bilobe  con- 
structed as  in  Fig.  58  from  the  ellipse  whose  minor  axis  is  2,  with  a 
trilobe  derived  as  in  Fig.  GO  from  the  one  whose  minor  axis  is  3,  etc., 
and  these  with  each  other  indifferently. 

119.  In  Fig.  69,  let  0  be  the  centre, 
A  and  C  the  foci,  and  HP  the  major 
axis  of  the  primitive  ellipse.     Draw 
CG  perpendicular  to  HP,  and  cut  it 
at  B  by  an  arc  with   centre  0  and 
radius  Off,  then  setting  off  BD,  DE, 
etc.,  each  equal  to  CB  the  semi-minor 
axis,  describe  arcs  about  0  through 
D,  E,  F,  etc.,  thus  determining  OK, 


FIG.  70. 

OM,  OR,  the  semi-major  axes  of  tho  ellipses  from  which  the  bilobe, 
trilobe,  and  quadrilobe  are  to  be  derived.  In  Fig.  70,  the  first  three 
of  a  series  are  shown  in  action,  the  unilobe,  or  primitive  ellipse, 
being  the  same  as  in  Fig.  69.  Since  the  difference  between  the 
greatest  and  the  least  radii  is  the  same  in  all  the  wheels,  being  equal 


60 


MAC   CORD'S   INTERCHANGEABLE  MULTILOBES. 


to  the  distance  between  the  foci,  it  will  be  seen  that  if  one  nmltilobe 
be  given,  the  original  ellipse  may  be  found  by  reversing  the  process, 
and  wheels  with  any  desired  numbers  of  lobes  constructed,  which  will 
roll  with  the  one  given. 

120.  The  other  method  of  constructing  a  set  of  lobed  wheels  capa- 
ble of  rolling  with  each  other 
indifferently,  was  devised  by 
the  author,  and  the  funda- 
mental curve  is  the  equiangular 
spiral.  The  process  is  illus- 
trated in  Fig.  71,  where  C  is 
the  pole  of  the  spiral  SMN. 
Drawing  through  C  any  right 
line,  cutting  the  curve  at  A 
and  By  the  portion  A  SB  is  one 
half  the  outline  of  a  unilobe 
which  will  roll  with  a  similar 
and  equal  one,  as  already 
shown.  In  order  to  construct 
a  bilobe  which  shall  roll  with 
this  unilobe,  it  is  necessary 
to  find  two  radiants  at  right 
angles  to  each  other,  whose 
difference  is  the  same  as  that 
between  CB  and  CA.  Pro- 
duce BA,  and  intersect  its 
prolongation  at  G  by  a  circle 
through  B,  with  centre  at  the 
pole  ;  and  draw  CF  perpendic- 
ular to  BA,  cutting  this  circle 


FIG.  71. 

at  F  and  the  spiral  at  H. 
Then 

and 


CB—CA  =  AG, 
CB—CH  =  HP. 


If,  now,  two  radiants  be  found,  which  are  to  CB  and  CH respect- 
ively, in  the  same  proportion  that  AG  bears  to  HF,  they  will  be 
perpendicular  to  each  other,  and  their  difference  will  be  equal  to  A  G. 

121.  This  may  be  done  graphically,  as  it  involves  merely  the  con- 
struction of  a  fourth  proportional  to  three  given  lines,  thus, 

A  G  :  HF  :  :  CB  :  CM, 
AG'.HF::CH:  OS. 


UNSYMMETRICAL   WILOBES. 


61 


The  arc  SBM  of  the  spiral  thus  found  is  one  quadrant  of  the  con- 
tour of  the  required  bilobe  ;  the  triangle  ADK  is  therefore  made  sim- 
ilar and  equal  to  MCS,  and  the  completion  of  the  wheel  requires  no 
explanation. 

For  the  trilobe,  draw  any  two  radiants  including  an  angle  of  GO0, 
ascertain  their  difference,  and,  comparing  it  with  A  G,  proceed  as  above 
to  increase  or  diminish  the  assumed  radiants,  as  the  case  may  be,  in 
the  same  proportion,  until  the  difference  is  equal  to  AG;  the  lengths 
thus  determined  will  be  those  of  the  greatest  and  least  radii  of  the 
trilobe ;  and  in  a  similar  manner  wheels  of  any  number  of  lobes  may 
be  constructed. 

122.  Since  in  any  system  of  interchangeable  multilobes,  the  differ- 
ence between  the  greatest  and  least  radii  must  be  constant,  while  their 
actual  lengths  vary  with  the  number  of  lobes,  the  limits  of  the  vari- 
ation in  the  velocity  ratio  will  not  be  the  same  for  different  pairs ;  and 
obviously  the  law  which  governs  the  rates  of  variation  as  well  as  the 
differences  between  those  limits,  depends  upon  the  peculiarities  of  the 
fundamental  curve — upon  the  eccentricity  of  the  primitive  ellipse  or 
the  obliquity  of  the  original  spiral. 

123.  Irregular  Lobed  Wheels. — The  non-circular  wheels  thus  far  de- 
scribed, whether  consisting  of  one  lobe  or  many,  are  symmetrical ;  and 
moreover,  the  whole  contour  of  each  is  made  up  of  curves  of  the  same 
kind.     Neither  of  these  things  is  so  of  necessity  ;.  and  as  illustrating 
the  possibilities  of  varied  motion  in  rolling  contact,  we  give  a  few  ex- 
amples of  irregular  wheels  composed  of  different  curves. 

In  Tig.  72,  let  O  and  D  be  the  poles  of  the  equal  and  similar  loga- 
rithmic spiral  arcs  ALB,  A  ME ; 
then  these  arcs  will  roll  together,  B 
and  E  meeting  at  G.  Now,  AB  may 
be  taken  as  the  major  axis,  and  C  as 
one  focus,  of  the  semi-ellipse  ANB, 
which  will  roll  with  the  equal  and 
similar  one  A  OE. 

In  each  revolution,  then,  the  veloc-  FIG.  72. 

ity  ratio  will  vary  between  the  reciprocal  limits 


CB 
CA> 


CA 


but  by  reason  of  the  different  natures  of  the  curves,  the  rates  of  vari- 
ation during  the  decrease,  will  not  be  the  converse  of  those  during 
the  increase. 


62  DISSIMILAE   UNILOBES. 

124.  In  Fig.  73,  let  C  be  the  pole  of  the  logarithmic  spiral  arc 
AFB.     With  centre  C,  and  radius  equal  to  %AB,  describe  an  arc  cut- 
ting the  spiral  at  F ';  draw  FA, 
FB,  FC,  also  FP  perpendicular 
to  AB,  and  on  BA  produced  set 
off  PL,  PM,  each  equal  to  FC. 
Then  because  CF-  CA  -CB- 
CF,  the  spiral  arcs  AF,  FB  are 

pToTra""  equal.     Consequently  the  chord 

FB  is  greater  than  the  chord  FA,  whence  PJS  is  greater  than  PA. 
That  is  to  say,  P  is  not  the  middle  point  of  AB  ;  but  it  is  the  middle 
point  of  ML  by  construction,  therefore  PL  is  greater  than  PA,  and 
PM  is  less  than  PB. 

The  spiral  arc  AFB,  turning  round  6Yas  a  fixed  centre,  will  roll 
with  an  equal  and  similar  one,  the  distance  between  their  centres  of 
motion  being  equal  to  A B.  Set  off  PR  =  PC',  then  R  and  C  are 
the  foci  of  the  ellipse  whose  major  axis  is  ML,  and  semi-minor  axis 
PF:  which  also  will  roll  with  an  equal  and  similar  one,  turning 
round  the  same  centre  C,  the  distance  between  centres  being  equal  to 
ML,  which  by  construction  is  equal  to  AB. 

125.  From  these  data  we  can  construct,  as  in  Fig.  74,  two  dissimilar 
unilobes,  whose  contours  are  com-  o  s 

posed  partly    of    the    spiral  and 

partly  of  the  elliptical  arcs.     We 

have  first  the  semi-ellipse  FLO,  of 

which   C  is  the  focus  and  P  the 

centre,  PL  being  the  semi-major 

axis  ;    this  rolls   with   the    equal  FIG.  74. 

semi-ellipse  GL8:  then  the  spiral  arcs  OB,  FB,  corresponding  to  FB 

of  the  preceding  diagram,  and  these  roll  with  the  equal  arcs  SA,  GA, 

which  correspond  to  FA  of  Fig.  73. 

During  the  revolution,  the  velocity  ratio  varies  between  the 
limits 

v         DL      v_        AD^ 
v'  =:  ~CL>     v'  '~~~  ~BC9 

which,  it  will  be  observed,  are  not  reciprocals,  as  in  the  unilobes 
previously  described. 

126.  In  Fig.  72,  the  wheels,  though  similar,  are  not  symmetrical ; 
in  Fig.  74  they  are  symmetrical  but  not.  similar  :    and  finally,  those 
represented  in  Fig.  75  are  neither  the  one  nor  the  other.     Above  the 


IRREGULAR  MULTILOBES. 


63 


line  of  centres,  each  wheel  is  formed  of  a  single  arc  of  a  logarithmic 

spiral,      A  MB     rolling     with 

ANE.      Below  that  line,    we 

have  BH,  which  rolls  with  EF, 

each  being  a  quadrant  of  the 

bilobe  derived,  as  in   Fig.    58, 

from  the  ellipse  whose   major 

axis  is  AB,  O  being  one  focus  ; 

AL,  which  rolls  with  AT,  these 

curves  being  parts   of  quadri-  FIG.  75. 

lobes  derived  in  the  same  way  from  the  same  ellipse  :  and  LH,  rolling 

with  IF,  these  two  being  equal  arcs  of  another  logarithmic  spiral. 

127.  It  may  be  added,  that  there  are  a  number  of   other  curves 
capable  of  rolling  in  contact  about  fixed  axes.     But  it  is  the  province 
of  the  mathematician  rather  than  of  the  mechanician  to  investigate 
their  properties,  which  do  not  admit  of  simple  geometrical  demon- 
stration ;  nor  aro  they  so  easily  constructed  and  adapted  to  use  in  me- 
chanical devices.    It -may  be  safely  said  that  in  general  the  number  of 
changes  in  the  velocity  ratio,  and  the  limits  of  its  variation,  are  of 
greater  importance  than  the  precise  law  according  to  which  it  varies 
between  those  limits  :  and  it  is  believed  that  the  principles  involved 
in  the  examples  already  given  will  be  found  to  meet  most  if  not  all 
the  requirements  of  praclical  mechanism. 

128,  These  various  curves,   whose  action  has  been   discussed   as 
though  they  were  mere  lines  rolling  with  each  other  in  their  own 
plane,  are  of  course  practically  to  be  considered  as  the  bases,  or  trans- 
verse sections,  of  cylindrical  surfaces,  tangent  to  each  other  along  an 
element.     Strictly,  therefore,  the  expression  that  <e  the  point  of  con- 
tact lies  always  on  the  line  of  centres,"  means  that  "the  element  of 
tangency  lies  always  in  the  plane  of  the  axes,"  which  is  one  condition, 
as  shown  in  Chap.  II.,  of  pure  rolling  contact  between  two  surfaces. 

And  equally  of  course,  the  consideration  of  these  surfaces  is  only 
the  step  preliminary  to  the  investigation  of  the  surfaces  of  teeth  to 
be  formed  upon  them  :  for,  the  motion  of  the  followers  being  com- 
pulsory only  so  long  as  the  contact  radii  of  the  driver  are  increasing, 
continuous  rotation  can  not  be  transmitted  by  means  of  these  rolling 
surfaces  themselves,  with  any  certainty  that  the  desired  velocity  ratio, 
whether  constant  or  varying,  will  be  exactly  maintained. 


CHAPTER   Y. 


ROTATION  BY  ROLLING   CONTACT,  AXES   NOT  PARALLEL. 


1.  AXES  INTERSECTING. — Velocity  Ratio  Constant  or  Varying.   The  Rolling  Cones, 

or  Pitch  Surfaces  of  Circular  and  Elliptical  Bevel  Wheels.     Conical  Lobed 
Wheels. 

2.  AXES  IN  DIFFERENT  PLANES. — Velocity  Ratio  Constant.     The  Rolling  Hyper- 

boloids,  or  Pitch  Surfaces  of  Skew-Bevel  Wheels. 

1.  Axes  Intersecting. 

129.  Velocity  Ratio  Constant.— It  follows  directly  from  the  deduc- 
tions at  the  conclusion  of  Chap.  II,  that  the  only  surfaces  capable  of 
rolling  in  line  contact  about  two  fixed  axes  which  intersect  each  other, 

are  cones,  the  common  vertex  being 
the  point  of  intersection,  and  the  com- 
mon element  lying  in  the  plane 
of  those  axes.  And  in  order  to 
maintain  a  constant  velocity  ratio, 
the  transverse  sections  of  these 
cones  must  be  circles.  Thus,  in  Pig. 
76,  let  AB,  AC,  be  the  axes,  and  AP 
the  common  element,  all  lying  in  the 
plane  of  the  paper  :  then  in  the  rota- 
tion indicated  by  the  arrows,  the 
motion  of  the  common  point  P  will 
Draw  PD  perpendicular  to  AB,  PE 


FIG.  76. 


be  perpendicular  to  the  paper, 
perpendicular  to  A  C,  and  let 


v    = 


V'    — 


ang. 


vel.  about  AB , 
"        "      AC; 


then  at  the  instant  we  shall  have 


PJ2 
PD 


THE   ROLLING   CONES.  65 

and  in  order  that  this  ratio  may  be  constant  as  required,  the  contact 
radii  PD,  PE,  must  be  constant  also.  That  is  to  say,  the  plane 
bases  of  the  cones,  PDF,  PEG,  perpendicular  to  the  axes,  must 
be  circles  ;  and  the  cones  themselves  are  generated  by  the  revolution 
of  the  common  element  AP  about  each  axis  in  succession. 

130.  These  then  are  the  pitch  surfaces  of  the  ordinary  conical  or 
bevel  wheels.     In  practice  it  is  of  course  sufficient  to  use  compara- 
tively thin  frusta  of  the  cones,  as  indicated  in  the  figure  :  in  the  great 
majority  of  cases  they  are  provided  with  teeth,  but  are  to  some  extent 
used  without,  constituting  what  is  called  Bevel  Friction  Gearing. 

As  above  deduced,  the  angular  velocities  are  always  inversely  pro- 
portional to  the  perpendiculars  let  fall  from  any  point  of  the  common 
element  upon  the  axes,  whatever  the  relative  positions  of  those  three 
lines.  But  it  is  easy  to  see  that  those  positions  may  be  very  different 
from  what  they  are  in  Fig.  76  ;  and  it  is  of  interest  to  consider  what 
results  may  follow  from  various  assigned  or  assumed  conditions. 

131.  First,  however,  it  may  be  remarked  that  in  practice  two  cases 
may  present  themselves  for  solution.     Thus,  the  axes  AB,  AC,  being 
given,  as  in  Fig.  76,  suppose  it  to  be  required  to  construct  a  pair  of 
wheels  so  proportioned  that  m  revolutions  about  AB  shall  produce 
n  revolutions  about  AC.    Draw  xx  parallel  to  AB,  at  a  distance  from 
it  measuring  n  parts  on  any  convenient  scale  of  equal  divisions,  and 
zz  parallel  to  AC,  at  a  distance  from  it  equal  to  m  parts  on  the  same 
scale.    These  two  lines  intersect  in  o ;  and  APo  is  evidently  the  com- 
mon element  of  a  pair  of  cones  which  will  roll  together  with  the  re- 
quired velocity  ratio. 

Or,  the  cone  APF  being  given,  it  may  be  required  to  find  another 
which  will  roll  with  it,  the  velocity  ratio  being  also  assigned  as  above. 
In  that  case  draw  xx  as  just  ex- 
plained, cutting  AP  in  oj  then 
about  o  as  centre,  describe  an  arc 
yy  with  radius  equal  to  m  parts  on 
the  scale,  and  the  axis  A  C  of  the 
required  cone  will  be  tangent  to 
this  arc  as  shown. 

132.  Now  in  Fig.  77,  let  the  axes 
AB,  CD,  cut  each  other  obliquely 
at  V,  the  angle  of  inclination  being 
given   and   the  velocity   ratio   as- 

.  -,  TV-          •  1  FlG-    77- 

signed.     Drawing  xx  and  zz  as  de- 
scribed in  the  preceding  article,  Vo  is  .determined  and  the  cones 
constructed. 


66 


BEVEL  WHEELS  IK   DOUBLE   PAIRS. 


It  is  to  be  noted  in  this  case,  that  these  surfaces  are  tangent  along 
the  single  line  oVl ;  and  that  if  one  shaft  be  carried  past  the  vertex, 
two  pairs  of  frusta  may  be  used  at  once,  cut  from  opposite  nappes  of 
the  cones,  as  E,  F,  and  G,  H.  These  two  pairs  may  be  equidistant  from 
the  vertex  and  therefore  exactly  alike,  as  shown ;  or  either  pair  may 
be  placed  nearer  the  vertex  if  desired,  as  they  roll  together  without 
any  interference,  with  the  same  velocity  ratio,  and  as  indicated  by  the 
arrows,  with  the  same  directional  relation. 

133.  In  this  instance  the  lines  xx,  zz,  are  drawn  within  the  acute 
angle  formed  by  the  intersecting  axes.     But  in  Fig.  78,  they  are 

,z  drawn  within  the  obtuse  an- 

gle, the  inclination  of  the 
axes  and  the  velocity  ratio 
being  the  same  as  before. 
The  result  is  a  different  pair 
of  cones  ;  the  velocity  ratio 
is  the  same  as  in  Fig.  77,  by 
construction  ;  but  the  direc- 
tional relation  is  changed. 

Also,  the  opposite  nappes 
of  these  cones  intersect  each 
other  in  such  a  way,  that 
though  under  the  condi- 
tions here  assumed  one  shaft  can  be  carried  past  the  vertex,  and  two 
pairs  of  frusta  employed  at  once  if  desired,  they  cannot  be  placed  at 
equal  distances  from  the  vertex  :  the  wheels,  therefore,  will  not  be 
alike,  but  one  pair  must  be  larger  than  the  other  in  order  to  avoid 
interference. 

134.  But  it  may  not  be  possible  to  use  two  pairs  at  once,  whatever 


FIG.  r,8. 


Fia.  79.  FIG.  80. 

their  distances  from  the  vertex.  Thus  in  Fig.  79,  the  cones  being 
constructed  so  as  to  produce  an  assigned  velocity  ratio  and  also  an 
assigned  directional  relation,  it  is  clear  that  neither  shaft  can  be  car- 
ried past  the  other  wheel. 


DOUBLE   TAKGEITCY   OF   COtfES. 


67 


Again,  it  is  to  be  observed  that  in  all  these  cases  the  tangency  is 
external :  and  it  obviously  always  will  be  when  the  common  element 
lies  within  the  acute  angle  formed  by  obliquely  intersecting  axes. 
But  when  it  lies  within  the  obtuse  angle,  it  may  happen,  as  in  Fig. 
80,  that  one  cone  shall  touch  the  other  internally.  Or,  the  common 
element  may  be  perpendicular  to  one  of  the  axes,  as  in  Fig.  81,  the 
pitch  cone  degenerating  into  a  plane. 

135.  Thus  far  the  axes  have  been  supposed  to  cut  each  other 
obliquely ;  and  whatever  the  forms  of  the  pitch  surfaces,  it  will  be 
noted  that  in  each  they  are  tangent  along  one  line  only.  We  have 

IQ 


FIG.  81. 


yet  to  consider  the  case  in  which  the  axes  are  perpendicular  to  each 
other,  as  in  practice  they  are,  more  frequently  than  otherwise.  And, 
as  shown  in  Fig.  82,  we  are  confronted  by  the  singular  circumstance 
that  the  pitch  cones  are  tangent  along  two  elements,  or,  mn.  The 
frustum  ^Tcan  then  roll  simultaneously  with  the  frusta  F  and  G, 


FIG.  83. 


cut  from  opposite  nappes  of  the  other  cone,  but  these  two  are  tangent 
to  H  along  different  lines,  and  consequently  they  will  turn  in  oppo- 
site directions.  Equal  and  opposite  frusta,  like  those  in  Fig.  77, 


68 


HOLLOW   CONE   WITH   DOUBLE   PAIR. 


cannot,  of  course,  be  simultaneously  employed  :  but  either  of  the 
shafts  can  be  carried  past  the  vertex,  and  the  unequal  pairs  of  wheels 
cut  from  the  opposite  nappes,  being  tangent  along  the  same  line,  will 
have  the  same  directional  relation.  And,  as  shown  also  in  Figs.  83 
and  84,  this  directional  relation  is  optional,  since  either  of  the  two 
common  elements  of  the  cones  may  be  selected  as  the  line  of  prag- 
matic contact. 

136.  In  the  case  of  internal  tangency,  equal  frusta  of  the  opposite 
nappes  of  the  hollo w  cone  may  be  combined  in  a  single  wheel,  as 

shown  in  Fig.  85.  Here  the  frusta 
F  and  H  are  formed  in  opposite  sides 
of  a  solid  ring,  which  is  carried  by 
arms  LL,  so  curved  as  to  avoid 
interference  with  the  wheel  G,  which 
engages  with  H,  and  is  fixed  on  the 
same  shaft  with  E,  engaging  with 
F.  By  this  arrangement,7"  the  forces 
transmitted  act  upon  the  large  wheel 
in  the  manner  of  a  couple,  in  oppo. 
site  directions,  on  opposite  sides  of 
and  equidistant  from  the  axis  -AB  • 
thus  relieving  the  bearings  from  side 
pressure. 

137.  We  find,  then,  that  whatever  the  angle  included  between  the 
axes,  it  is  always  possible  to  construct  two  pairs  of  cones,  rolling  to- 
gether with  the  same  velocity  ratio,  but  having  different  directional 
relations,  and  we  are  at  liberty  to  employ  whichever  may  best  serve 
the  purpose  to  be  accomplished. 

The  cones  can  always  be  determined  when  the  axes,  velocity  ratio 
and  directional  relation  are  given,  by  the  process  described  in  (131). 

Another  method,  differing  slightly  in  detail,  is  shown  in  Figs.  86 
and  87,  which  also  illustrate  the  following  property,  viz  :  that  if  dis- 
tances be  set  off  on  the  axes  from  the  point  of  intersection,  directly 
proportional  to  the  angular  velocities,  and  the  parallelogram'  com- 
pleted of  which  these  are  the  adjacent  sides,  the  common  element 
will  lie  in  the  direction  of  the  diagonal. 

Thus  let  v'  the  angular  velocity  about  AB'  be  represented  by  VL, 
and  v',  that  about  CG,  by  VM,  then  VP  will  be  the  common  element. 

For,  draw  PD  perpendicular  ioAB,  PE  perpendicular  to  CG,  and 
join  DE.  Now  VP  divides  the  parallelogram  ML  into  two  triangles 


FIG.  85. 


Which  was  first  suggested  by  Mr.  0.  A.  Benton. 


PARALLELOGRAM  OF  ANGULAR  YELOC 


F  I  V  E^R  SIT 

£x/r    o:Er      *-tV 
which  are  similar  to  each  other  ;  and  also  to  DPE,  because  a  circle 

will  go  round  VEPD,  in  which  the  angles  DEP,  D  VP,  stand  on  th( 


FIG.  86.  FIG.  87. 

same  arc,  and  are  therefore  equal.  For  a  like  reason  the  angle  EDP 
is  equal  to  EVP9  which  is  equal  to  VPL.  Then  if  VP  be  the  com- 
mon element,  we  have 


PE 
PD 


VL 
LP 


VL 

~VM 


If  the  angle  between  the  axes  be  acute,  it  may  be  better  to  set  off 
VL  representing  v,  and  then  to  draw  LP  parallel  to  CG,  and  of  a 
length  representing  v'  on  the  same  scale  ;  thus  avoiding  the  uncer- 
tainty arising  from  attempting  to  locate  P  by  an  acute  intersection. 

In  these  two  diagrams  the  inclination  of  the  axes,  and  the  Telocity 
ratio,  are  the  same,  but  the  directional  relations  are  unlike.  Conse- 
quently VM has  the  same  ratio  to  VL  in  both  cases,  but  is  set  off  so  that 
in  one  diagram  the  angle  MVL  is  acute,  while  in  the  ot*her  it  is  obtuse. 

138.  Velocity  Ratio  Varying,— In  Fig.  88  let  E,  F,  be  two  pins 
fixed  in  the  sphere  whose  centre  is  0; 
let  EFP  be  a  loop  of  fine  inextensi- 
ble  thread,  passing  around  the  pins 
and  a  marking  point  at  P.  Then  as 
P  is  moved  along,  the  thread  being 
kept  always  taut  and  in  contact  with 
the  sphere,  it  will  trace  the  curve 
APBHj  which  \sthesphericalettipse. 

Evidently,  EF,  EP  and  FP  will 
always  be  arcs  of  great  circles  ;  and 
since  the  part  EF  of  the  thread  is 
always  the  same,  and  always  idle, 


70  THE   SPHERICAL  ELLIPSE. 

while  the  total  length  is  invariable,  this  curve  may  be  defined  as  the 
locus  of  the  vertices  of  a  series  of  spherical  triangles,  of  which  the  base 
is  common  and  the  sum  of  the  other  two  sides  constant.  This  sum 
will  be  equal  to  the  major  axis  AB  ;  now  let  AD  be  the  major  axis  of 
an  equal  and  similar  curve  tangent  to  the  first  at  A,  of  which  G  and 
H  are  the  foci  :  then  BD  will  be  a  continuous  arc  of  a  great  circle, 
and  EHw\\\  be  equal  to  AB  or  AD. 

Next,  let  these  two  curves  be  taken  as  bases  of  cones  whose  com- 
mon vertex  is  0,  the  centre  of  the  sphere  :  then  they  will  be  tangent 
to  each  other  along  the  element  OA,  and  they  will  roll  in  contact 
about  OE  and.  OH  as  fixed  axes.  For,  let  AP,  AR  be  any  equal  arcs 
of  the  two  spherical  ellipses  ;  then  we  shall  have 
» 

PE=RG,  PF=RH, 

whence, 

PE  +  RH  =  PE  +  PF  =  EH-, 

consequently  the  points  P  and  R  will  meet  on  the  arc  DAB,  or  in 
other  words,  the  common  element  will  always  lie  in  the  plane  of  the 
axes. 

139.  AVe  have,  then,  a  pair  of  non-circular  cones,  capable  of  rolling 
in  contact  about  fixed  intersecting  axes,  with  a  varying  velocity  ratio, 
and  forming  the  pitch  surfaces  of  what  may  be  properly  called  Ellip- 
tical Bevel  Wheels,  since  their  bases,  as  thus  constructed,  are  spheri- 
cal ellipses.     But  though  these  curves  are  easily  traced  upon  the  sur- 
face of  a  sphere  in  the  mechanical  manner  above  described,  it  will  be 
found  practically  more  convenient  to  employ  others,  which  may  be 
constructed  as  follows.     Referring  to  Fig.  88,  it  is  seen  that  the  sides 
of  the  spherical  triangles  are  the  measures  of  the  faces  of  a  series  of 
trihedral  angles  whose  common  vertex  is    0.      Join    0  with  the 
centre  C  of  the  spherical  ellipse  ;  then  a  plane  perpendicular  to  00 
will  cut  the  edges  of  these  pyramids  in  points  readily  determined,  and 
the  problem  of  finding  the  plane  base  of  the  cone  resolves  itself  into 
the  construction  of  a  number  of  trihedral  angles,  of  which  one  face  is 
common  and  the  sum  of  the  other  two  constant.     The  base  thus  found 
will  not  be  a  true  ellipse,  but  it  will  closely  resemble  one,  and  evi- 
dently will  be  symmetrical  about  two  lines  at  right  angles  to  each 
other. 

140.  The  processes  of  constructing  one  of  the  trihedral  angles  in  ac- 
cordance with  the  above  conditions,  and  of  determining  by  means  of 
it  a  point  in  the  required  curve,  are  illustrated  in  Figs.  89  and  90. 


CONSTRUCTION   OF   TRIHEDRAL   ANGLES. 


71 


FIG.  90. 


The  lines  OE,  OC,  OF,  of  Fig.  89  correspond  to  those  similarly  lettered 
in  Fig.  88  ;  they  all  lie  in  the  hori- 
zontal plane,  and  OE  is  perpendicu- 
lar to  GL  the  ground  line.  EOF, 
then,  is  the  common  face  of  the  tri- 
hedral angles,  bisected  by  OC.  As- 
sume ono  of  the  two  remaining  faces, 
as  EOA,  producing  OA  to  cut  GL 
in  A.  Subtract  this  assumed  face 
from  the  given  constant  sura,  thus 
determining  the  third,  which  lay  off 
in  the  horizontal  plane  as  FOB,  mak- 
ing OB  equal  to  OA. 

Draw  BH  perpendicular  to  OF, 
produce  it  to  cut  GL  in  p  ;  erect 
pp'  perpendicular  to  GL,  and  of 
a  length  equal  to  the  altitude  of  a  right-angled  triangle  of  which  HP 
is  the  base  and  HB  the  hypothenuse  :  then  p  is  the  horizontal  and  p' 
is  the  vertical  projection  of  the  point  where  B  will  fall  in  the  vertical 
plane  if  OB  is  revolved  about  OF. 

Also,  if  OA  be  revolved  about  OE,  A  will  describe  a  circle  in  the 
vertical  plane;  which  will  pass  through  p',  because  OA  =  OB; 
hence  OP  is  the  horizontal  and  Ep'  is  the  vertical  projection  of  the 
third  edge  of  the  trihedral  angle  required. 

141.  Now  let  J/jVbe  a  plane  perpendicular  to  OC ';  it  will  cut  this 
third  edge  in  a  point  whose  horizontal  projection  is  r,  and  r's'  will 
be  the  vertical  projection  of  a  perpendicular  let  fall  from  this  point 
to  the  horizontal  plane. 

In  Fig.  90,  which  is  a  projection  upon  this  new  vertical  plane  MN, 
the  horizontal  plane  is  seen  as  G'L,  and  the  point  r  appears  at  R, 
the  distance  RS  being  equal  to  r's'  of  Fig.  89  :  the  line  0(7  appears 
as  the  point  C,  and  i  and  &,  the  points  in  which  OE  and.  OF  are  cut 
by  MN,  are  seen  as  /  and  K. 

In  this  projection  the  required  plane  base  of  the  cone  is  seen  in  its 
true  form  ;  as  many  points'  as  may  be  necessary  to  determine  its  con- 
tour with  precision,  being  found  exactly  as  R  was,  by  the  construc- 
tion of  other  trihedral  angles. 

142.  In  Fig.  91,  O.E'and  05"  are  the  fixed  axes  of  a  pair  of  frusta 
cut  from  cones  of  this  description ;    the  extremities  of  the  major 
axes  of  the  bases  are  shown  in  contact  at  A,  the  plane  of  the  paper 
in  this  figure  corresponding  to  the  plane  ODAB  of  Fig.  88. 

The    action  .  is    obviously   akin  to    that  of    two    ellipses   rolling 


72 


ELLIPTICAL   BEVEL   WHEELS. 


together  in  their  own    plane ;    letting  fall    upon    the   fixed   axes 

the   perpendiculars   AM,    AN, 
the  velocity  ratio  in  the  present 

AM 

position  is  —  ^  >  and  at  the  end 

of  a  half  revolution  it  will  be 

AN 

AM' 

This  combination  also  resem- 
bles the  elliptical  spur-wheels 
in  the  respect  that  if  these  con- 
ical frusta  overhang  their  bear- 
ings, as  here  shown,  a  link,  L, 
may  be  used  to  connect  what 
it  is  proper  to  call  their  free  foci  : 
the  axes  of  the  pins  by  which 
this  link  is  pivoted  to  the  wheels, 
of  course  converging  in  0  the 
FIO.  91.  common  vertex. 

143.  Any  reciprocal  limits  may  be  assigned  to  the  velocity  ratio ; 

and  if  the  angle  between  the  axes  be  also  given,  the  pitch  cones  may 

be  constructed  as  follows. 


Let  the  assigned  values  of  the  velocity  ratio  be  - 


m    n 


-  5  and  let  OE, 
m 


OH,  Fig.  92,  be  the  given  axes. 
Draw  xx,  zz,  parallel  to  the 
axes,  and  at  distances  from 
them  which  are  to  each  other 
in  the  ratio  of  m  to  n ;  these 
lines  intersect  at  P.  Draw  PO, 
and  lay  off  the  angle  POB 
equal  to  EOH-,  draw  OCbisect- 
ing  POB,  and  from  any  point 
A  of  PO,  draw  AB  perpendic- 
ular to  00,  cutting  OB  in  B. 
Then  A  B  may  be  taken  as  the 
major  axis  of  the  plane  base  of 
the  cone,  which  from  these  data  can  now  be  constructed  in  the 
manner  previously  explained ;  for  making  the  angle  BOF  equal  to 
A  OE,  we  have  EOF  the  common  face,  and  A  OB  the  constant  sum 
of  the  other  two  faces,  to  be  used  in  determining  the  trihedral  angles, 
whose  edges  are  the  elements  of  the  cone. 


ELLIPTICAL   BEVEL   WHEELS. 


73 


FIG.  93. 


144.  In  Fig.  93,  the  inclination  of  the  axes  is  the  same  as  in  Fig. 
92,  but  the  lines  xx,  zz,  are  so  drawn   that  their  intersection  P  falls 

within  the  obtuse  instead  of  in  the 
acute  angle  :  the  result  being  that 
the  directional  relation  is  different. 

From  the  very  nature  of  the  case, 
an  engaging  pair  of  elliptical  bevel 
wheels  must  be  equal  and  similar, 
and  always  in  external  contact. 
And  it  will  be  seen  that,  since  the 
angle  subtended  at  the  vertex  by  the 
major  axis  of  the  base,  is  equal  to 
that  included  between  the  fixed  axes 
of  rotation,  neither  cone  can  degen- 
erate into  a  plane,  until  that  angle 
becomes  180°,  and  the  axes  coincide  : 
much  less  can  either  be  hollow. 

We  have  then  always  the  choice  between  two  pairs  of  wheels,  hav- 
ing different  directional  relations  but  the  same  action  in  respect  to 
the  velocity  ratio. 

If  the  axes  be  perpendicular  to  each  other,  as  in  Fig.  94,  the  pitch 
cones  will,  like  the  circular 
ones  under  the  same  condi- 
tion, have  two  common  ele- 
ments, either  of  which  may 
be  selected  as  the  line  of  prag- 
matic contact. 

145.  Conical  Lobed  Wheels. 
— From  these  elliptical  bevel 
wheels,  it  is  possible  to  con- 
struct   conical    wheels    with 
various  numbers  of  lobes,  by 
a  process  of  contraction  or  ex- 
pansion of  angles  analogous  to  that  applied  to  the  plane  ellipses. 

Let  the  elements  cut  from  one  of  a  pair  of  these  pitch  cones  by  a 
series  of  radiating  planes  through  the  fixed  axis,  bo  revolved  about  that 
axis  until  the  dihedral  angles  between  the  planes  are  expanded  or  con- 
tracted in  any  desired  proportion.  Any  part  of  the  elliptical  cone 
may  be  thus  transformed  into  a  new  conical  surface,  which  will  roll 
in  contact  with  the  one  formed  by  treating  in  like  manner  the  corre- 
sponding portion  of  its  rolling  mate. 

The  bases  of  these  lobed  cones  will  be  similar  to  the  outlines  of  the 


74  COKICAL  LOBED   WHEELS. 

multilobes  derived  from  the  plane  ellipses.  By  applying  the  process 
of  contraction  to  one  half  the  perimeters  of  the  spherical  ellipses,  pairs 
of  similar  wheels,  with  any  desired  number  of  lobes,  may  be  con- 
structed analogous  in  form  and  action  to  those  shown  in  Figs.  59 
and  61. 

146.  It  is  also  practicable  to  construct  conical  lobed  wheels  which 
will  roll  together  in  dissimilar  pairs,  unilobe  with  bilobe,  trilobe  with 
quadrilobe,  and  so  on.  Eef erring  to  Fig.  88,  it  will  be  seen  that  the 
dihedral  angles  PFE,  1?HG,  are  equal.  Hence  if  the  spherical  ellipses 
be  such,  for  example,  that  the  dihedral  angles  RGA,  RHG,  are  to 
each  other  as  3  is  to  2,  the  former  may  be  contracted  to  90°,  and  the 
latter  will  by  contraction  in  the  same  proportion  be  reduced  to  60°  ; 
the  result  being  the  formation  of  a  bilobe  rolling  with  a  trilobe,  the 
bases  resembling  those  of  Fig.  62. 

It  is  sufficient  merely  to  state  that  by  constructing  a  series  of  spher- 
ical triangles,  as  was  done  with  the  plane  triangles  in  Fig.  GG,  a  spher- 
ical curve,  analogous  to  the  one  there  shown,  may  be  drawn,  by  the 
aid  of  which  we  can  determine  the  limits  of  the  eccentricity  of  the 
spherical  ellipses  from  which  it  is  possible  to  derive  such  dissimilar 
pairs  with  assigned  numbers  of  lobes.  It  may  be  said,  and  no  doubt 
with  truth,  that  the  difficulty  of  making  such  wheels  would  prevent 
their  being  used  under  any  ordinary  circumstances.  But  it  is  equally 
true  that  only  extraordinary  conditions  would  require  them  to  be  used  : 
and  should  cases  occur  in  which  it  would  be  desirable,  it  is  worthy  of 
note  that  very  many  cf  the  combinations  of  non-circular  cylindrical 
wheels  described  in  the  preceding  chapter,  may  be  replaced  by  combi- 
nations of  conical  wheels  nearly  identical  in  their  action. 

147.  Of  these,  it  is  probable  that  the  ones  derived  as  above  suggested 
from  the  spherical  ellipse,  would  be  found  practically  preferable,  as 
being  the  ones  most  easily  constructed,  and  also  as  allowing  a  wider 
range  in  the  selection  of  the  limits  of  variation  in  the  velocity  ratio. 
But  the  equiangular  spiral  has  also  its  spherical  analogue,  which  may 
be  made  the  base  of  a  conical  surface,  capable  of  rolling  in  contact 
with  an  equal  and  similar   one,  the  fixed   axes  of   rotation  passing 
through  the  poles  of  the  bases  and  the  centre  of  the  sphere  upon  which 
they  lie. 

If  from  any  point  on  the  sphere  arcs  whose  lengths  arc  in  geometri- 
cal progression  be  set  off  successively  on  equidistant  meridians  passing 
through  the  point,  the  curve  drawn  through  the  extremities  of  these 
arcs  will  be  the  one  required. 

148,  This  will  be  readily  seen  by  the  aid  of  Fig.  95.     In  the  side 
view  let  0(7  be  a  vertical  radius  of  the  sphere,  and  AB  a  plane  tan- 


NON-CIRCULAR   CONES. 


75 


gent  to  it  at  C.  Let  C  in  this  plane  be  the  pole  of  the  logarithmic 
spiral  unilohe  shown  in  dotted  outline,  as  AMNm  the  horizontal  pro- 
jection. 

The  radiants  of  this  spiral  which  include  equal  angles,  are  in  geomet- 
rical progression  ;  and  planes  passing  through  them  and  also  through 
OC,  cut  equidistant  meridians 
from  the  sphere.  On  each  me- 
ridian set  off  from  C  an  arc 
equal  in  length  to  the  corre- 
sponding radiant  of  the  spiral : 
the  result  will  be  the  spherical 
curve  shown  in  full  lines ;  the 
vertical  projection  being  A'P'B'. 

Now  let  A' RE  be  the  vertical 
projection  of  an  equal  and  simi- 
lar curve,  of  which  D  is  the 
pole  :  the  vertical  plane  being 
that  of  the  great  circle  contain- 
ing OC,  OD,  and  also  the  arcs 
A'C'B',  ADE.  It  is  then  obvi- 
ous that  these  curves  will  be  tan- 
gent to  each  other  at  A;  also, 
that  if  A'P,  AR  be  equal  arcs 
of  the  two,  then  P  and  R  will 
meet  in  the  plane  of  the  axes,  if 
the  curves  revolve  about  OC  and 
OD  respectively,  as  shown  by 
the  arrows.  For  the  angular 
distances  of  these  points  from 
that  plane  are  the  same  as  in  the  original  spirals,  which  are  known 
to  roll  together,  and  by  construction  the  sum  of  the  arcs  CP,  DR,  is 
equal  to  the  arc  CD. 

149,  It  will  be  seen,  then,  that  if  these  curves  be  made  the  direc- 
trices of  cones  whose  common  vertex  is  0  the  centre  of  the  sphere, 
those  surfaces  will  work  in  rolling  contact  about  OC  and  OD  as  fixe*d 
axes  ;  and  that  the  velocity  ratio  will  vary  from  maximum  to  mini- 
mum and  back  again  once  in  each  revolution,  this  combination  being 
analogous  to  that  of  the  two  logarithmic  spiral  unilobes  shown  in  Fig. 
53. 

In  fact  the  plane  original  ANN  is  identical  with  one  of  those 
unilobes.  But  the  actions  of  the  two  combinations,  though  very  sim- 
ilar, are  not  identical.  Draw  AS,  AT,  perpendicular  to  0(7  and 


Fia.  95. 


76  NOX-CIRCULAR  COXES. 

OD,  the  limiting  values  of  the  velocity  ratio  in  Fig.  95  are 

A_8     AT , 
A'T9  A'S' 
while  in  Fig.  53  they  are 

AC     _  arc  A'C 
~BC9       arcB'C' 
and 

BC_     _  arc  B'C 
AC9       arc  A'C' 

By  comparing  in  a  similar  manner  the  values  of  the  velocity  ratio 
for  intermediate  positions,  it  will  be  found  that  neither  the  limits  nor 
the  laws  of  variation  are  precisely  alike  in  the  two  cases. 

150.  If  the  angle  COD  between  the  axes  be  given,  and  the  values  of 
the  reciprocal  limits  of  the  velocity  ratio  be  assigned,  the  cones  may 
be  constructed,  by  processes  closely  resembling  those  employed  in  pre- 
ceding cases. 

Thus,  it  is  evident  that  OA'  must  pass  through  the  intersection  of 
two  lines  xx,  zz,  drawn  parallel  to  OC  and  OD  respectively,  and  at 
distances  from  them  which  are  to  each  other  in  the  proportion  of  A'S 
to  A'T-,  then  CB'  =  A'D,  and  the  arcs  A'C  and  CB'  being  rectified 
give  AC  and  CB,  whence  the  original  spiral  A  MB  may  be  recon- 
structed, and  the  spherical  curves  derived  from  it  as  just  described. 

Like  preceding  combinations,  too,  this  might  be  discussed,  with 
reference  to  the  effects  of  dividing  the  obtuse  instead  of  the  acute 
angles,  of  placing  the  axes  of  rotation  at  right  angles,  and  in  short,  of 
assuming  any  special  conditions.  It  will  also  readily  be  seen  that  the 
bilobes,  trilobes,  etc.,  derived  from  the  equiangular  spiral  and  rolling 
in  contact  about  parallel  axes,  have  their  analogues  in  conical  multi- 
lobes  whose  axes  intersect.  All  these  matters,  however,  we  shall  leave 
the  reader,  if  so.  disposed,  to  pursue  farther  at  his  leisure  :  the  exam- 
ples already  given  illustrating  sufficiently  the  principles  involved  in 
the  construction  of  this  class  of  combinations,  which,  it  is  proper  to 
add  in  conclusion,  we  believe  to  be  a  new  one. 

2.  Axes  in  Different  Planes. 

151.  Velocity  Eatio  Constant. — If  a  right  line  revolve  about  an  axis 
in  a  different  plane,  the  surface  generated  is  the  hyperboloid  of  revo- 
lution.    Any  normal  to  this  surface  will  intersect  the  axis ;  it  will 
also  be  perpendicular  to  both  generatrices  through  the  point  of  nor- 
malcy, since  these  two  elements  determine  the  tangent  plane. 

If  then  a  series  of  normals  be  drawn  through  different  points  of  the 


AXES   IK   DIFFERENT   PLANES. 


77 


revolving  line,  they  will  lie  in  planes  perpendicular  to  that  line,  and 
therefore  parallel  to  each  other.  They  are,  consequently,  elements 
of  one  generation  of  a  hyperbolic  paraboloid,  of  which  the  directrices 
are  the  axis  and  the  revolving  line,  and  the  plane  directer  is  perpen- 
dicular to  the  latter. 

Any  plane  parallel  to  those  directrices  will  therefore  cut  the  series 
of  normals  in  points  which  will  lie  in  one  right  line,  an  element  of 
the  second  generation  of  the  hyperbolic  paraboloid. 

Now  the  line  thus  determined  may  be  taken  as  a  new  axis  ;  and  by 
revolving  around  it,  the  same  line  which  generated  the  first  hyper- 
boloid  will  generate  another.  These  two  surfaces  of  revolution,  hav- 
ing, at  every  point  of  a  common  rectilinear  element,  a  common  nor- 
mal and  a  common  tangent  plane,  will  be  tangent  to  each  other  all 
along  that  element. 

152.  Since  these  hyperboloids  are  warped  surfaces,  perfect  rolling 
contact  between  them,  as  has  been  shown,  is  not  possible  under  any 
circumstances  whatever. 

But  they  are  capable  of  rotating 
in  contact  about  fixed  axes,  with  a 
constant  velocity  ratio;  and  the 
sliding  between  them  is  quite  dif- 
ferent from  that  between  two  tan- 
gent cones  or  cylinders  whose  peri- 
metral  velocities  are  not  the  same, 
in  the  respect  that  it  is  wholly  in 
the  direction  of  the  common  ele- 
ment. And  these  hyperboloids,  like 
the  cylinders  and  the  cones,  are 
practically  used  as  the  pitch  sur- 
faces of  toothed  wheels.  In  view  of 
these  facts,  it  is  proper  to  consider 
their  action  in  this  place,  notwith- 
standing the  imperfection  in  their 
rolling. 

153.  The  form  of  the  surface,  and 
the  manner  of  constructing  it,  are 
shown  in  Fig.  96.     As  the  inclined 
line  AB  revolves  about  the  vertical 
axis,  every  point  in  it  describes  a 

circle  in  a  horizontal  plane,  whose  radius  is  seen  in  its  true  length  in 
the  horizontal  projection.  In  that  projection,  the  axis  appears  as  the 
point  C,  and  CD  is  the  common  perpendicular  of  the  axis  and  the 


78 


THE   ROLLING   HYPERBOLOIDS. 


revolving  line.  Then  the  point  D  describes  the  circle  of  the  gorge, 
EDE;  A  describes  the  circle  of  the  upper  base,  FAF;  B,  that  of 
the  lower  base,  GBG,  in  this  case  equal  to  FAF,  because  A  and  B 
are  equidistant  from  D  :  and  any  intermediate  point  L  describes  a 
circle  00,  whose  radius  is  OL.  In  this  way  any  number  of  points  in 
the  meridian  outline  GOJEJFmay  be  determined. 

The  same  surface  may  be  generated  by  the  revolution  about  the 
same  axis,  of  another  right  line  MDN\  MNan&AB  having  the 
same  horizontal  projection,  and  being  equally  inclined  to  the  plane  of 
rotation,  but  in  opposite  directions. 

For  the  paths  of  A  and  M,  also  those  of  B  and  N,  coincide,  and  D 
is  common  to  both  lines :  consequently  any  two  points,  one  on  each 
line,  equidistant  from  D,  as  for  example  L  and  P,  will  describe  the 
same  circle. 

154.  Through  any  point  of  the  surface,  then,  two  rectilinear  ele- 
ments, or  companion  generatrices,  may  be  drawn  ;  whose  projections 
on  a  plane  perpendicular  to  the  axis  will  be  tangent  to  that  of  the 
gorge  circle  on  the  same  plane. 

Thus  AH,  tangent  at  K  in  the  horizontal  projection .  to  the  circle 
EDE,  is  the  second  generatrix  through  A.  In  this  figure,  the  verti- 
cal plane  contains  the  axis ;  AH 
pierces  this  plane  at  /,  as  deter- 
mined from  the  horizontal  projec- 
tion :  and  in  the  vertical  projec- 
tion AH  is  tangent  at  /  to  the 
hyperbolic  outline,  of  which  E  is 
the  vertex,  and  AB  and  MN  are 
the  asymptotes. 

These  two  lines,  AB  and  AH, 
determine  the  plane  tangent  to  the 
hyperboloid  at  A  ;  the  horizontal 
trace  of  this  plane  is  therefore 
parallel  to  BH ;  and  its  vertical 
trace  is  parallel  to  AB,  since  that 
line  is  parallel  to  the  vertical  plane. 
Consequently  AC,  perpendicular 
to  BH,  is  the  horizontal,  and  AR 
perpendicular  to  AB,  is  the  verti- 
cal, projection  of  the  normal  to 
the  surface  at  A. 

155,  Now  in  Fig.  97,  two  hy- 
perboloids  are  shown  in  contact,  the  axes  A  C  and  FE,  and  the  com- 


DETERMINATION   OF   VELOCITY   KATIO.  79 

mon  element  AB,  being  parallel  to  the  vertical  plane.  In  the  hori- 
zontal projection  the  vertical  axis  appears  as  the  point  C',  the  in- 
clined one  as  F'E',  and  the  common  element  as  AB1,  which  at  A' 
cuts  the  common  perpendicular  of  the  axes,  here  seen  in  its  true 
length  as  (7T,  into  segments  A  'C',  C'l',  which  are  the  radii  of  the 
two  gorge  circles.  Also  C'B'E'  is  the  horizontal,  and  CBE  is  the 
vertical,  projection  of  the  common  normal  to  the  surfaces  at  B.  In 
the  vertical  projection,  the  upper  bases  and  the  gorge  circles  will 
appear  as  lines  perpendicular  to  the  axes  :  the  latter  intersecting  at  A, 
and  the  former,  which  in  the  figure  passes  through  B,  cutting  the  axes 
at  M  and  N. 

156.  Let  these  hyperboloids  revolve  about  their  axes  in  the  direc- 
tions indicated  by  the  arrows.     The  point  A  of  the  inclined  surface 
will  at  the  instant  move  in  the  direction  of  the  tangent  to  the  gorge 
circle  at  that  point :  which  being  parallel  to  the  vertical  plane,  this 
motion  may  be  represented  in  magnitude  and  direction  by  AHm  the 
vertical  projection  :  and  it  may  be  resolved  into  the  components,  AS 
in  the  direction  of  AB,  and  AO  perpendicular  to  that  line. 

Considering  the  inclined  hyperboloid  as  the  driver,  the  rotation  of 
the  vertical  one  will  not  be  compulsory,  because  the  contact  radii  are 
constant. 

But  the  common  element  may  here,  as  in  the  cases  of  the  cylinders 
and  the  cones,  be  regarded  as  the  line  of  contact  of  teeth  formed  upon 
the  two  surfaces,  so  that  AO  would  be  the  normal,  and  AS  the  tan- 
gential component  of  AH ;  because  AB  would  necessarily  be  a  line 
of  the  plane  tangent  to  both  teeth  at  A.  Therefore  the  resultant 
motion  of  that  point  of  the  vertical  hyperboloid  (whose  direction  is 
tangent  to  the  larger  gorge  circle),  must  be  of  such  magnitude,  AL, 
as  to  have  the  same  normal  component  A  0,  the  other  component 
being  A  Tin  the  line  AB. 

157.  Now  let  v  =  ang.  vel.  about  FE ' , 

v'  =    "      "        "     AC; 
then 


a). 

A'F 


But  from  similar  triangles  AHL,   ACE, 

AH  _     AE_ 
~AL  ~~     AC  ' 


80  DETERMINATION   OF  VELOCITY   RATIO. 


and  by  the  principles  of  projection, 
Substituting  in  (I), 


A^_(7  _  I?Cr_  _  BC 
AT'  ^  B'E1  ~  BE 


v'       BE      AC 
From  similar  triangles  ABC,  ABM, 

Afi  _  AB  m 
BE~~  B~M' 

and  from  similar  triangles  ABC,  ABN, 


_    BN 
AC  ~~  ~AB  ' 

Substituting  in  (2),  we  have,  finally,  for  the  velocity  ratio, 

v_       BN 

v'  ~  BM  ' 

That  is  to  say,  the  angular  velocities  are  to  each  other  in  the  in- 
verse ratio,  not  of  the  perpendiculars  let  fall  from  any  point  of  the 
common  element  upon  the  axes,  but  of  the  projections  of  those  per- 
pendiculars upon  a  plane  parallel  to  both  axes  and  the  common 
element. 

158.  Supposing  the  axes  to  be  given  in  position,  then,  and  the 
velocity  ratio  assigned,  the  inclination  of  the  common  element  is 
found  exactly  as  in  the  case  of  intersecting  axes  :  two  lines,  xx,  zz, 
are  drawn  parallel  to  ^C'and  AE,  at  distances  from  them  which  are 
to  each  other  in  the  inverse  ratio  of  the  angular  velocities  about  those 
axes  respectively  ;  and  AB  must  pass  through  their  intersection  P. 
Then  drawing  CE  perpendicular  to  AB,  it  is  the  vertical  projection 
of  the  normal,  therefore  the  horizontal  projection  of  E  must  be  at  E' 
on  the  horizontal  projection  of  FE  ;  and  projecting  B  to  B'  on  C'E', 
we  draw  B'A'  parallel  to  F'E',  thus  determining  the  horizontal  pro- 
jection of  the  common  element  and  also  the  radii  of  the  gorge  circles, 
AC'  and  A  I'  :  whence  the  surfaces  may  be  constructed  as  before 
described.  Or,  if  for  example  the  vertical  surface  be  given,  and  it 
be  required  to  find  the  other,  the  velocity  ratio  being  also  assigned  : 
then,  knowing  j?^and  its  ratio  to  BM,  the  length  of  the  latter  may 
be  found,  and  with  that  as  radius,  an  arc  is  described  -about  B  as 


HYPERBOLOIDS   IXTEEtfALLY   TAKGENT. 


81 


centre.  The  vertical  projection  FE  of  the  second  axis  is  drawn 
through  A,  tangent  to  this  arc.  The  vertical  projection  of  the  nor- 
mal at  By  is  CB  perpendicular  to  AB,  and  it  cuts  FE  in  E,  whose 
horizontal  projection  must  therefore  be  at  E'  on  the  prolongation  of 
C'B'y  and  F'E'  parallel  to  A ' B'  is  the  horizontal  projection  ol  the 
axis  of  the  required  surface. 

159.  In  Fig.  97,  the  common  element  lies  between  the  axes,  and 
the  two  surfaces  touch  other  exter- 
nally.    But  if  one  be  larger  than 

the  other,  the  smaller  one  may  be 
placed  within  it,  and  that  in  such 
a  position  as  to  touch  the  larger 
along  a  line  of  the  concave  surface. 
Thus,  in  Fig.  98,  the  same  pair  of 
hyperboloids  as  in  the  preceding 
figures,  are  shown  in  this  new  rela- 
tion :  the  demonstration  of  the  ve- 
locity ratio  applies,  it  will  be  seen, 
without  any  change  whatever,  but 
the  directional  relation  in  the  two 
cases  is  different.  The  surfaces  are 
tangent  along  an  element  of  the 
same  generation  as  before,  but  it 
now  lies  on  the  same  side  of  both 
axes. 

It  will  be  observed  that  in  the 
case  in  which  the  vertical  hyper- 
boloid  and  the  velocity  ratio  are 
given,  FAE  is  drawn  tangent  to 
the  same  circle,  described  about  B 
with  radius  BM,  but  on  the  side  opposite  to  that  selected  in  Fig.  97. 

Also,  that  in  the  other  case,  where  the  two  axes  and  the  velocity 
ratio  are  given,  the  lines  xx,  zz,  are  so 'drawn  that  their  intersection 
P  lies  in  the  obtuse  angle  formed  by  the  axes,  instead  of  in  the  acute 
one  as  in  the  preceding  figure. 

160,  In  the  above  determination  of  the  velocity  ratio,  the  motions 
of  the  coincident  points  of  the  two  gorge  circles  were  selected  merely 
because  they  were  the  most  convenient  ones  for  the  purpose.     It  is 
evident  that  the  angular  velocity  of  every  point  in  either  surface  about 
its  axis  must  be  the  same.     The  motion  of  the  line  AB  of  the  vertical 
surface  is  one  of  revolution  about  AC ';  and  AL  being  the  linear  ve- 
locity of  Ay  every  point  in  AB  must  have  a  component  motion  along 


FIG.  98. 


82  BATE   OF   SLIDIKG. — TRANSVERSE   OBLIQUITY. 

that  line  equal  to  A  T,  and  in  the  same  direction,  as  shown  in  Figs.  8 
and  9.  And  in  revolving  about  FE,  the  motion  of  every  point  of 
AB  must  have  a  component  in  the  direction  of  and  equal  to  AS, 
since  AH  is  the  linear  velocity  of  the  point  A  in  the  inclined  hyper- 
boloid. 

The  action  of  the  combination,  therefore,  consists  in  rolling,  com- 
bined with  a  sliding  in  the  line  of  the  common  element,  which  is  rep- 
resented by  ST,  the  sum  or  the  difference,  as  the  case  may  be,  of 
AS  and.  AT. 

The  rate  of  sliding,  then,  is  at  any  instant  the  same  at  every  point 
of  contact ;  but  as  the  linear  velocity  of  each  point  depends  upon  its 
distance  from  the  axis  of  the  surface  upon  which  it  lies,  it  is  evident 
that  the  proportion  of  the  sliding  component  to  the  perimetral  veloc- 
ity will  be  greatest  at  the  gorge,  and  diminish  as  the  point  under  con- 
sideration recedes  therefrom. 

161.  Although  in  Figs.  97  and  98,  the  hyperboloids,  in  order  to  save 
space,  are  limited  by  the  gorge  planes,  it  is  obvious  that  they  may  be 
extended  to  any  distance  on  both  sides  of  those  planes,  and  will  be 
tangent  from  end  to  end.     But  practically,  as  in  bevel  gearing,  com- 
paratively thin  frusta  only  of  the  pitch  surfaces  are  used  ;  and  their 
location  is  optional  within  certain  limits  in  some  special  cases  here- 
after to  be  noted.     Thus  in  Fig.  99  the  surfaces  are  in  contact  all 
along  the  line  mn  ;  and  we  may  use  either  of  the  three  pairs  of  frusta 
A  and  B,  C  and  Z>,  E  and  F,  or  any  two  or  all  .three  pairs  at  once  : 
and  the  steadiness  of  the  motion  will  practically  be  greater  when  two 
pairs  equidistant  from  the  gorge  planes  are  employed. 

162.  Transverse  Obliquity. — The  plane  tangent  to  the  hypefboloid  at 
.any  point  is  determined  (154)  by  the  companion  generatrices  through 
the  point.     If  a  meridian  plane  be  passed  through  the  same  point,  it 
will  cut  the  tangent  plane  in  a  line  tangent  to  the  hyperbolic  outline 
;at  that  point  :  and  the  angle  included  between  this  line  and  either 
generatrix  is  the  measure  of  what  is  called  the  transverse  obliquity, 
or  slceiv,  of  the  teeth  which  mifst  be  used  in  practice. 

As  before  mentioned,  teeth  which  work  in  line  contact  ultimately 
reduce  to  rectilinear  elements  of  their  pitch  surfaces.  But  when  they 
are  of  sensible  magnitude,  there  is  one  instant  in  the  action  of  each 
engaging  pair,  when  their  line  of  tangency  coincides  with  the  common 
element  of  the  pitch  surfaces.  When  those  are  cylinders  or  cones,  the 
common  element  lies  in  the  plane  of  the  axes  :  this  is  impossible  in 
the  case  of  the  rolling  hyperboloids,  but  the  less  the  departure  from 
that  condition,  the  more  advantageous  will  be  the  action.  And  this 
transverse  obliquity,  which  measures  the  inclination  of  the  generatrix 


SIMILARITY   OP   HYPEKBOLOIDS   TO    CO^ES.  83 

to  the  meridian  plane  at  any  point,  is  obviously  greatest  at  the  gorge, 
and  diminishes  as  the  element  is  extended,  which  is  another  reason 
for  employing,  when  the  circumstances  of  the  case  admit  of  it,  frusta 
at  some  distance  from  the  gorge  planes. 

Keferring  to  Fig.  96,  it  will  be  seen  that  the  transverse  obliquity  at 
the  gorge  is  measured  by  the  angle  TDA  in  the  vertical  projection. 
That  at  A  is  half  the  angle  between  AB  and  AH\  to  construct  it,  de- 
scribe with  radius  the  true  length  of  AB  (seen  in  the  vertical  projec- 
tion), arcs  about  B  and  H  in  the  horizontal  projection,  where  BH  ap- 
pears in  its  true  length.  These  arcs  intersect  at  8  on  the  prolonga- 
tion of  CA  ;  and  C8B  or  CSHis  the  required  angle. 

163.  Now  since,  when  the  axes  are  given  in  position,  the  angle 
between  their  projections  on  a  plane  parallel  to  both  is  divided,  in 
order  to  produce  any  assigned  velocity  ratio,  precisely  as  though  the 
axes  intersected  and  the  pitch  surfaces  were  cones,  it  will  be  seen  that 
in  every  case  the  problem  admits  of  two  solutions.     That  is  to  say ; 
with  any  given  pair  of  axes,  it  is  possible  to  construct  two  pairs  of 
hyperboloids,  having  the  same  velocity  ratio,  but  different  directional 
relations. 

The  cone  is,  in  fact,  but  the  limiting  form  of  the  hyperboloid,  in 
which  the  radius  of  the  gorge  circle  becomes  zero  :  and  all  the  pecu- 
liarities which  have  been  pointed  out  as  resulting  from  special  condi- 
tions with  intersecting  axes,  will  be  found  to  have  almost  their  exact 
counterparts  under  analogous  circumstances  when  the  axes  lie  in 
different  planes. 

164.  Thus,  when  the  projections  of  the  axes  cut  each  other  oh- 
liquely,  we  may  divide  the  acute  angle,  as  in  Fig.  97,  and  the  resulting 
hyperboloids  are  externally  tangent :  a  comparison  of  Fig.  99,  which 
represents  the  same  pair  of  surfaces,  with  Fig.  77,  will  clearly  show 
the  close  resemblance  between  the  two  combinations. 

If  the  obtuse  angle  be  divided  instead,  one  hyperboloid  may  be 
internally  tangent  to  the  other,  as  in  Fig.  98  ;  which  condition  of 
things  is  at  once  seen  to  correspond  to  the  case  of  a  cone  rolling 
within  a  hollow  one,  as  shown  in  Fig.  79. 

But  whether  the  axes  intersect  or  not,  it  does  not  follow  that  the 
phenomenon  of  internal  tangency  will  always  occur  when  the  obtuse 
angle  between  their  projections  is  selected.  This,  in  relation  to  coni- 
cal surfaces,  was  illustrated  in  Figs.  78  and  79  :  and  the  truth  of  it 
in  regard  to  the  surfaces  now  under  consideration  will  be  evident  on 
examination  of  Fig..  100,  in  which  the  relative  positions  of  the  axes, 
and  the  velocity  ratio,  are  precisely  the  same  as  in  Figs.  97  and  99, 
the  only  difference  being  in  the  directional  relation.  This,  as  above 


84 


RESULTS   OF   DIVIDING   THE   OBTUSE   ANGLE. 


stated,  is  due  to  the  division  of  the  obtuse  instead  of  the  acute  angle ; 
but  the  tangency  of  the  pitch  surface  is  still  external,  as  it  always  will 
be  if  the  acute  angle  be  divided. 

165.  And  this  figure  calls  attention  to  another  point  of  similarity 
between  the  cone  and  the  hyperboloid.     The  latter,  although  a  sur- 


FIG.  99. 


FIG.  100. 


face  of  one  nappe,  is  divided  by  the  gorge  plane  into  two  parts,  whose 
relations  are  very  like  those  of  the  opposite  nappes  of  the  cone.  In 
Fig.  100,  it  will  be  observed  that  the  shaded  frusta,  £f  and  D,  might 
be  extended  to  the  gorge  planes.  Not  beyond,  however,  because  the 
extension  G  of  the  inclined  surface  will  intersect  the  frustum  D  of 
the  vertical  one,  and  F,  the  extension  of  the  latter  surface,  will  inter- 
sect H :  a  condition  of  things  very  similar  to  that  in  Fig.  78. 

Equal  and  opposite  pairs  of  frusta,  then,  cannot  be  used  in  this 
case  :  nor  yet  can  we  employ  a  pair  like  the  central  one  of  Fig.  99, 
of  which  the  mid-planes  are  the  gorge  circles.  But  if  H  and  D  be 
very  near  the  gorges,  and  very  thin,  it  will  be  possible  to  carry  the 
inclined  shaft  past  D,  and  to  use  at  the  same  time  another  pair, 
G  and  F\  if  they  be  sufficiently  far  from  the  gorge  planes  to  clear 
the  first  pair. 

166.  Again,  ifc  may  happen  that  when  the  projection  of  the  com- 
mon element  lies  in  the  obtuse  angle,  it  shall  be  perpendicular  to  the 
projection  of  one  of  the  axes.  This  case,  shown  in  Fig.  101,  presents 
the  remarkable  feature  that  the  pitch  hyperboloids  retain  their  limit- 
ing forms,  the  one  remaining  a  cone,  the  other  a  plane. 


DEGENERATED   HYPERBOLOIDS.  (i  83 

£V  „          OJET 

It  will  be  seen  that  the  common  element  AB 
inclined  axis  FE,  by  revolving  around  which  it  generates  the  cone, 
ABL  :   while  in  revolving 
about  the  vertical  axis,  the 
points  A  and  B  describe  the 
circles  whose  radii  are  C'A'9 
C'B',  in  the  horizontal  pro- 
jection. 

The  determination  of  the 
velocity  ratio  will  be  most 
conveniently  made  by  con- 
sidering the  motions  about 
the  axes  of  the  two  points 
which  fall   together  at  B. 
The   motion   of  the  point 
which  belongs  to  the  plane 
hyperboloid,  may  be  repre- 
sented by  B'H,  perpendicu-  FIG.  101. 
lar  to  C'B'  in  the  horizontal  projection  ;  and  it  may  bo  resolved  into 
the  components,  B'Tm  the  line  of  the  common  element,  and  B' 0 
perpendicular  to  it. 

Now  B'O  is  tangent  to  the  circle  described  about  the  inclined  axis 
by  the  point  which  belongs  to  the  cone,  and  therefore  represents  its 
resultant  motion,  which  has  no  component  along  A'H'. 

167.  Then  as  before,  let 


and  we  have 


v  =  ang.  vei.  about  inclined  axis, 
v'  =    "       "        "      vertical    "  , 


B'C' 


v'      B'H       BN 


0)- 


B'C'  ) 
But  from  similar  triangles  B' OH,  B'A'C',  we  have, 

B'O       A'B'         AB 


B'H      B'C' 


since  A'B  =  AB  : 


Substituting  in  (1), 


v'    "  BN 


86 


DOUBLE   TANGENCY   OF   HYPERBOLOIDS. 


As  in  the  other  cases,  then,  the  velocity  ratio  depends  entirely  upon 
the  projections  of  the  axes  and  the  common  element  on  a  plane  par- 
allel to  all  three,  and  the  singular  circumstance  results  that  the  angu- 
lar velocity  of  the  cone  is  not  affected  by  the  lateral  separation  of  the- 
axes  or  the  resulting  variation  in  the  diameter  of  the  perforated  disc 
,  with  which  it  works  in  contact. 

The  sliding  component  B'T  de- 
pends wholly  upon  the  revolution  of 
AB  about  the  vertical  axis,  and 
since  it  must  be  the  same  at  every 
point  of  the  moving  line,  its  ratio 
to  the  linear  velocity  is  greatest  at 
the  vertex  of  the  cone,  diminishing 
as  we  recede  from  that  point. 

168.  Finally,  in  the  case  in  which 
the  projections  of  the  axes  intersect 
each  other  perpendicularly,  as  in 
Fig.  102,  it  is  seen  that  the  pitch 
hyperboloids  are  tangent  to  each 
other  along  two  elements,  as  were 
the  cones  under  the  same  circum- 
stances. 

FIG.  102.  These  elements,  mn,  op,  are  neces- 

sarily the  comparion  generatrices,  and  evidently  tend  to  establish 
the  same  velocity  ratio,  but  different  directional  relations.     Either  of 

them  may  be  made  the  line  of  prag- 
matic contact,  and  at  the  limit,  as 
shown  in  Fig.  103,  a  frustum  B  of 
one  hyperboloid  may  work  in  con- 
tact with  two  frusta,  A  and  C,  on 
opposite  sides  of  the  gorge  plane  of 
the  other.  But  this  combination 
could  practically  be  used  only  as  an 
arrangement  of  friction  gearing,  the 
wheels  being  parts  of  the  pitch  sur- 
faces only,  for  teeth  of  sensible  magnitude  must  be  disposed  in  the 
direction  of  a  generatrix,  common  to  the  engaging  frusta ;  now  B 
and  C  are  tangent  along  one  line,  B  and  A  along  another,  and  the 
teeth  of  B  cannot  slope  both  ways  at  once. 

169,  Evidently  this  double  tangency  prevents  in  this  case  also  the 
use  of  frusta  whose  mid-planes  are  the  gorge  circles,  as  well  as  of  equal 
and  opposite  pairs.  But  pairs  on  opposite  sides  of  the  gorge  planes 


FIG.  103. 


SKEW   FRICTION   GEARING.  87 

may  be  used,  having  the  same  common  element,  and  therefore  the 
same  directional  relation,  if  they  be  placed  at  different  distances  from 
those  planes.  Either  line  of  contact  maybe  selected,  so  that  as  in  the 
case  of  the  cones  the  directional  relation  is  optional.  The  resulting 
combinations  are  shown  in  Figs.  104  and  105  ;  comparing  these  with 


FIG.  104.  t          FIG.  105. 

Figs.  83  and  84,  the  analogy  between  the  cone  and  the  hyperboloid, 
considered  as  the  pitch  surfaces  of  wheels,  will  be  most  clearly  seen. 

170.  It  may  be  remarked  that  frusta  of  these  hyperboloids  can  be 
employed  in  the  manner  of  friction  gearing.  When  this  is  to  be  done 
the  frusta  are  preferably  placed  at  some  distance  from  the  gorge 
planes  ;  for  the  curvature  of  the  hyperbola  diminishes  so  rapidly  as  it 
recedes  from  the  vertex,  that  it  very  soon  becomes  almost  inappreciable. 
Consequently,  if  at  the  mid-planes  of  the  frusta  thus  located  cones  be 
drawn  tangent  to  the  hyperboloids,  frusta  of  these  cones  may  practi- 
cally be  employed.  It  is  hardly  necessary  to  repeat  that  since  the 
transmission  of  the  motion  depends  wholly  upon  the  adhesion  be- 
tween the  surfaces  in  contact,  no  absolute  dependence  can  be  placed 
upon  the  constancy  of  the  velocity  ratio,  and  it  may  be  noted  that, 
since  in  order  to  secure  such  adhesion,  it  is  necessary  to  press  the  sur- 
faces together  with  considerable  force,  an  amount  of  friction  is  thus 
caused  in  the  bearings^  so  great  that  it  may  be  questioned  whether 
such  gearing  has  in  many  cases  any  advantage  other  than  that  of 
original  simplicity  and  possibly  in  freedom  from  noise  at  high  veloc- 
ities, over  toothed  wheel- work ;  which  Jatter  we  will  now  proceed  to 
discuss. 


CHAPTEE   VI. 


1.  CLASSIFICATION  OF  TOOTHED  GEARING. 

2.  SPUR  WHEELS.—  Epicycloidal  Teeth,  Outside  Gear,  Generation  of  the  Outline. — 

Pitch,  Angles  of  action,  Backlash,  Clearance,  Approaching  and  Receding  Ac- 
tion.— Interchangeable  Wheels.— Size  of  Describing  Circle. — Rack  and  Pinion. 

3.  INSIDE  SPUR  GEARING. — Intermediate  Describing  Circle. — Limiting  Diameters 

of  Generating  Circles. 


1.   Tlie  Different  Varieties  of  Gearing  Classified. 

171.  It  has  been  pointed  out,  that  when  a  definite  Telocity  ratio  is  to 
be  maintained,  the  pitch  surfaces  previously  discussed  are  unsuitable 
for  the  transmission  of  rotation,  on  account  of  their  liability  to  slip 
upon  each  other  ;  and  we  propose  now  to  consider  the  forms  of  the 
teeth  which  in  practice  must  be  employed  in  order  to  prevent  this 
slipping. 

But  it  is  desirable  first  to  gain  clear  ideas  of  the  general  nature  of 
the  various  kinds  of  toothed  wheels  in  use,  and  of  the  peculiarities 
upon  which  their  classification  is  based. 

172.  Not  only  may  the  axes  of  a  pair  of  engaging  wheels  have  dif- 
ferent relative  positions,  but  the  teeth  themselves  may  be  of  different 
kinds,  and  act  upon  each  other  in  different  ways  ;  for  example,  the 
mode  of  action  of  a  pair  of  screw-wheels  is  quite  dissimilar  in  its  in- 
trinsic nature  from  that  of  a  pair  of  skew-bevel  wheels,  although  the 
relative  positions  of  the  axes  may  be  the  same  in  each  case.     There 
are,  in  consequence,  six  varieties  or  classes  of  toothed  gearing  to  be 
met  with  in  practice,  viz  : 

1.  Spur  Gearing.        4.  Twisted  Gearing. 

2.  Bevel  Gearing.        5.  Screw  Gearing. 

3.  Skew  Gearing.        G.  Face  -Gearing. 

173.  Eegarding  this  matter  from  a  new  point  of  view,  it  is  seen  that 
the  teeth  of  engaging  wheels  act  upon  each  other  by  contact  what- 
ever their  number.     If  then  that  number  be  indefinitely  increased,  the 
size  being  correspondingly  diminished,  the  teeth  will  ultimately  be- 


CLASSIFICATION   OF   GEARING. 


89 


come,  in  general,  mere  lines,  or  elements  of  surfaces  in  contact.  The 
relative  motions  of  these  surfaces  will  be  the  same  as  those  of  the 
wheels  from  which  they  are  thus  derived,  their  forms  and  disposition 
depending  on  the  nature  of  the  class  of  gearing  to  which  those  wheels 
originally  belonged  :  these  are  the  pitch  surfaces,  whose  action  in 
some  combinations  has  already  been  investigated.  For  the  purpose 
of  comparison,  and  to  illustrate  the  distinctive  features,  a  pair 
of  wheels  of  each  class,  and  also  their  pitch  surfaces,  are  shown  in 
Figs.  106  to  111,  inclusive. 

174.  In  the  first  three  classes,  the  engaging  teeth,  which  are  bounded 


FlG.  106,     SPUll   GEARING. 


FlG.  107,   BEVEL  GEARING. 

by  ruled  surfaces,  touch  each  other  along  right  lines,  and  by  the  process 
above  indicated  they  are  reduced  to  rectilinear  elements  of  the  pitch 
surfaces,  which  by  the  mode  of  derivation  must  be  tangent  all  along  a 
right  line.  As  already  stated,  the  axes  of  spur  wheels  are  parallel, 
and  their  pitch  surfaces  are  cylinders  ;  the  axes  of  bevel  wheels  inter- 
sect, and  their  pitch  surfaces  are  cones  whose  common  vertex  is  the 
point  of  intersection  :  the  axes  of  skew  wheels  lie  in  different  planes, 
and  their  pitch  surfaces  are  hyperboloids. 

175.  Let  us  now  suppose  one  of  a  pair  of  engaging  circular  wheels, 


90 


CLASSIFICATION   OF   GEAKING. 


belonging  to  either  of  these  three  classes,  to  be  uniformly  twisted  on 
its  axis,  each  successive  transverse  plane  being  rotated  through  a 
greater  angle  than  the  preceding  one  ;  then  the  other  wheel  of  the 
pair  will  receive  a  corresponding  twist,  as  will  be  readily  understood 
by  the  aid  of  Fig.  109.  It  will  hereafter  be  shown  that  the  twisted 


FlG.  109,   TWISTED  GEARING. 

| 

wheels  thus  formed  will  gear  together  as  well  as  before,  and  in  sub- 
stantially the  same  manner.  The  teeth  are  now  distorted  into  sur- 
faces of  a  helicoidal  nature,  and  by  the  above  process  of  indefinite  sub- 
division, they  reduce  to  helical  lines. 

But  it  is  to  be  noted  that  these  lines  lie  upon  surfaces  which  are 
tangent  along  a  right  line,  whether  the  axes  are  parallel,  intersecting, 
or  neither.  And  it  will  also  be  seen  that  whatever  of  screw-like 
action  may  be  involved  in  their  motions,  tends  only  to  cause  pressure 
in  the  direction  of  the  common  element  of  the  pitch  surfaces,  and 
has  nothing  to  do  with  the  transmission  of  rotation. 

176.  In  these  respects  there  is  a  marked  distinction  between  these 
twisted  wheels,  and  those  belonging  to  the  next  class,  of  Screw  Gear- 
ing properly  so  called ;  although  they  are  frequently  confounded  with 
each  other.  In  all  the  latter,  the  teeth,  it  is  true,  are  also  of  helicoidal 


TWISTED   AND   SCREW   WHEELS   CONTRASTED. 


91 


form,  and  reduce  to  helical  lines  :  but  these  helices  lie  upon  cylinders 
whose  axes  are  in  different  planes,  and  the  pitch  surfaces  therefore 
touch  each  other  in  a  single  point  only.  Moreover,  as  illustrated  by 
the  familiar  combination  of  the  "worm  and  wheel,"  it  is  the  screw- 
like  action  alone  of  one  wheel  upon  the  other,  by  which  the  rotation  is 
transmitted  in  that  class  of  gearing. 

177.  Face  Gearing  is  not  much  used  in  modern  machinery  ;  the 
name  is  derived  from 

the  fact  that  the 
wheels  were  usually 
formed  with  teeth 
consisting  of  turned 
pins  projecting  from 
the  faces  of  cir- 
cular' disks,  as  shown 
in  Fig.  Ill  :  a  mode 
of  construction  well 
adapted  to  wooden 
mill  work,  and  to 
that  only.  In  the 
case  illustrated  here, 
the  axes  are  perpen- 
dicular to  each  other ; 
but  turned  pins  may 
be  inserted  in  other  surfaces  than  planes,  and  in  this  way  such  wheels 
can  be  made  to  work*  together  when  the  axes  have  different  relative 
positions.  All  these  may  be  properly  said  to  belong  to  the  same 
class  ;  of  which  the  distinguishing  features  are,  that  whatever  the 
relation  of  the  axes  or  the  general  forms  of  the  wheels,  the  teeth  are 
circular  in  their  transverse  sections,  touch  each  other  in  a  single 
point,  and  ultimately  become  points  in  the  circumferences  of  circles 
which  are  in  contact.  The  reason  of  this  last  peculiarity  is,  thafc  an 
increase  of  the  number  involves  a  diminution  of  the  length  as  well  as 
of  the  diameters  of  the  teeth,  so  that  at  the  limit  they  vanish  alto- 
gether :  whereas,  in  the  other  classes  of  gearing,  the  length  of  the 
teeth  is  not  affected  by  any  variation  in  the  height  or  thickness, 
and  they  reduce  to  lines.  Face  wheels,  then,  have  no  pitch 
surfaces  properly  so  called,  although  in  constructing  them,  surfaces 
of  some  kind  must  be  provided  in  which  to  secure  the  teeth  or  pins. 

178.  The  following  table  exhibits  in  a  convenient  manner  the  pecu- 
liar features  of  the  different  kinds  of  gearing  above  mentioned  :  the 
teeth,  of  whose  linear  elements  the  forms  are  given  in  the  last  column, 


FlO.    Ill,   FACE   GEARING. 


TABLE   OF   DISTINCTIVE   FEATUEES. 


being  supposed  to  be  of  sensible  magnitude,  in  order  that  the  circular 
sections  of  those  in  the  sixth  class  may  be  kept  in  view. 


CLASS   OP  GEARING. 

RELATIVE  POSITIONS 
OF  AXES. 

PITCH  SURFACES. 

ELEMENTS   OF  TEETH. 

1.  SPUR. 

Parallel. 

Cylinders. 

Rectilinear. 

2.  BEVEL. 

Intersecting. 

Cones. 

Rectilinear. 

3.  SKEW. 

In  Different  Planes. 

Hyperbole-ids. 

Rectilinear. 

4.  TWISTED. 

Any. 

Either. 

Helical. 

5.  SCREW. 

In  Different  Planes. 

Cylinders. 

Helical. 

C.  FACE. 

Any. 

None. 

Circular. 

2.  The  Teeth  of  Spur  Wheels. 

Epicycloidal  System. 
179.    Generation  of  the  Tooth  Outline,— In  Fig.  112,  let  C,  D  be  the 


T  FIG.  112. 

centres  of  the  pitch  circles  LM9  EN.     Tangent  to  these  at  A,  is  a 
smaller  circle  whose  centre  is  0.     Suppose  all  the  centres  to  be  fixed, 


GENERATION   OF  TOOTH-OUTLINE.  93 

then  the  three  circles  can  move  in  rolling  contact,  with  equal  linear 
velocities.  Set  off  from  A  the  three  equal  arcs  AB,  AE,  A  P.  Suppose 
a  marking  point  fixed  originally  at  A,  in  the  circumference  of  the 
smaller  circle ;  then  while  this  travels  to  P,  it  will  trace,  with  refer- 
ence to  RX,  the  curve  BP,  and  with  reference  to  LM,  the  curve  EP. 
Now  the  relative  motions  of  the  circles  are  precisely  the  same  as 
though  the  smaller  one,  which  carries  the  marking  point,  had  rolled 
upon  the  outside  of  RN,  and  upon  the  inside  of  LM,  regarding  these 
two  as  fixed  base  lines  :  the  curves  tire,  therefore,  an  epicycloid  and 
a  hypocycloid  respectively  ;  and  AP  is  their  common  normal  at  P, 
because  on  whichever  of  the  two  fixed  lines  we  regard  the  small  circle 
to  be  rolling  at  the  instant,  the  point  of  contact  A  is  the  instantaneous 
axis  (73),  so  that  the  motion  of  P  in  cither  curve  is  perpendicular  to 
A  P.  If  the  tracing  point  go  on  to  P,  the  arcs  AP',  AE',  AB'9 
being  equal,  the  resulting  curves  B'P',  E'P,  are,  clearly,  but  exten- 
sions of  the  first  pair,  and  AP'  is  their  common  normal. 

180.  We  perceive,  then,  that  the  curves  thus  simultaneously  gener- 
ated are  tangent  to  each  other  at  some  point,  throughout  the  gener- 
ation ;  that  the  point  of  tangency  is  always  in  the  describing  circle  : 
and  that  the  common  normal  always  passes  through  the  fixed  point 
A9  upon  the  line  of  centres. 

Consequently  these  curves  are  correct  outlines  for  parts,  at  least,  of 
teeth  ;  if  the  curved  lever  CE  turn,  as  shown  by  the  dotted  arrow,  it  will 
drive  the  other  before  it,  the  point  of  contact  following  the  arc  P' PA, 
until  E'  and  B'  meet  at  A,  and  as  the  common  normal  always  cuts  the 
line  of  centres  at  the  same  point,  the  velocity  ratio  will  be  constant. 

181.  It  is  to  be  noted,  that  this  condition  would  have  been  satisfied, 
had  the  tracing  point  been  fixed,  not  in  the  circumference  of  the  de- 
scribing circle,  but  at  a  greater  or  less  distance  from  its  centre.     And 
it  will  also  be  readily  perceived,  that  the  tracing  point  need  not  be 
carried  by  a  circle  at  all ;  any  other  describing  curve  might  have  been 
used,  provided  that  it  were  capable  of  rolling  in  contact  with  the 
pitch  circles;  the  point  of  contact  would  not  travel  in  a  path  coin- 
ciding with  the  describing  curve,  but  the  generated  curves  would 
always  have  a  point  of  tangency,  and  the  common  normal  would 
always  have  passed  through  A.      And  conversely,  any  two  curves 
of  which  these  two  things  are  true,  can  be  generated  in  the  manner 
above  described.     In  general,  then  :  The  tooth-outlines  which  act  in 
contact,  must  be  such  as  can  be  simultaneously  traced  upon  the  planes 
of  rotation  of  the  two  wheels  while  in  action,  by  a  marking  point 
which  is  carried  by  a  describing  curve  moving  in  rolling  contact  with 
both  pitch  circles. 


94  LAYIKG   OUT  THE   TEETH. 

By  using  various  describing  curves,  then,  an  infinite  number  of 
tooth-outlines  may  be  generated,  all  of  which  geometrically  satisfy 
the  conditions.  But  many  of  them  are  of  impracticable  forms  ;  of 
those  which  are  not,  none  have  been  more  extensively  employed  than 
the  Epicycloid  and  the  Involute,  to  which  therefore  we  shall  at  pres- 
ent confine  our  attention  ;  and  we  now  proceed  to  the  practical  oper- 
ations of  "laying  out  the  teeth  "  of  a  pair  of  wheels  in  outside  gear. 

182.  Circular  Pitch, — Supposing  the  distance  between  the  centres  of 
a  pair  of  wheels  to  be  given,  and  the  velocity  ratio  to  be  assigned  ;  the 
first  step  is  to  divide  the  line  of  centres  into  segments  having  the 
given  ratio,  thus  determining  the  radii  of  the  pitch  circles.     The  cir- 
cumference of  each  circle  is  next  to  be  divided  into  as  many  equal 
parts  as  its  wheel  is  to  have  teeth. 

The  pitch  of  the  teeth  is  the  length  of  the  circular  arc  measuring 
one  of  these  subdivisions  ;  or,  in  other  words,  it  is  the  distance  meas- 
ured on  the  pitch  circle,  occupied  by  a  tooth  and  a  space.  This  pitch 
arc,  it  is  obvious,  must  be  the  same  on  each  wheel ;  although  the 
teeth  may  be  smaller,  and  the  spaces  larger,  on  one  wheel  than  upon 
the  other.  The  numbers  of  the  subdivisions,  then,  are  proportional 
to  the  diameters  of  the  pitch  circles;  and  a  fractional  tooth  being 
impossible,  the  pitch  must  be  an  aliquot  part  of  each  circumference. 

The  pitch  as  above  defined  is  sometimes  called  the  Circular  pitch, 
in  distinction  from  what  is  called  the  Diametral  pitch,  which  will  be 
explained  hereafter. 

183.  Face  and  Flank. — The  part  of  a  tooth-outline  which  lies  out- 
side its  pitch-circle,  as  B'P'  in  Fig.  112,  is  technically  called  ihefaoe 
of   the  tooth  ;   and  the  part  which  lies  within  the  pitch  circle,   as 
E'P'  in  the  same  figure,  is  called  \\\Q  flank.     Usually,  each  tooth  has 
both  ;  but  wheels  can  be  made,  and  sometimes  used  to  great  advan- 
tage, in  which  one  of  a  pair  has  faces  only,  the  other  one  having  only 
flanks  ;  we  will  consider  this  case  first. 

184.  Arc  and  Angle  of  Action. — The  angle  through  which  a  wheel 
turns,  while  one  of  its  teeth  is  in  contact  with  the  engaging  tooth  of 
another  wheel,  is  called  the  angle  of  action :  and  the  arc  of  the  pitch 
circle  by  which  it  is  measured,  is  called  the  arc  of  action.     The  latter 
must,  evidently,  be  at  least  equal  to  the  pitch  arc,  in  order  that  each 
tooth  may  continue  in  gear  until  the  next  one  begins  to  act;  and  it 
ought  in  practice  to  be  considerably  greater. 

185.  Backlash. — The  size  of  the  tooth  depends,  partially  at  least, 
upon  the  pitch,  since,  as  above  stated,  the  pitch  arc  is  to  be  divided 
between  a  tooth  and  a  space.    Practically,  of  course,  the  teeth  of  both 
wheels  are  made  of  the  same  thickness ;  hence,  were  perfect  work- 


A   LIMITING    CASE. 


95 


man  ship  attainable,  the  tooth  and  the  space  might  be  made  exactly 
equal.  But  since  it  is  not,  the  space  must  be  made  a  little  wider  than 
the  tooth  ;  the  difference  is  called  backlash,  and  should  be  as  small  as 
it  is  practicable  to  make  it.  For  our  present  purposes  we  may  neg- 
lect it  altogether,  and  make  the  thickness  of  the  tooth  just  one  half 
the  pitch. 

A  Pair  of  Wheels. — Limiting  Case. 
186,  In  Fig.  113  the  pitch  and  describing  circles  being  drawn  as  in 


II 


the  preceding  figure,  let  AB,  AE,  be  the  pitch  arcs,  and  AP  an  equal 
arc  on  the  describing  circle.  Then  iheface  for  the  tooth  of  RN  can- 
not be  less  than  BP,  since  if  made  of  just  that  height,  as  shown,  con- 
tact is  ending  at  P  at  the  very  instant  when  the  next  tooth  begins  to 
act  at  A.  Bisect  AB  in  H,  which  gives  the  thickness  of  the  tooth  ; 
and  draw  through  Ha,  reversed  face  similar  to  BP. 

The  conditions  are  purposely  so  chosen  that  this  reversed  face  passes 
through  P  :  the  case  is,  therefore,  a  barely  possible  one,  the  tooth 
being  pointed,  and  just  high  enough  to  make  the  angle  of  action  equal 
to  the  pitch  angle. 

We  founjl  that  the  face  must  be  of  the  height  BP,  in  order  to  se- 
cure this  angle  of  action  :  drawing  PD,  which  cuts  the  pitch  circle 
in  0,  we  see  that  in  this  case  BG  is  just  half  the  thickness  of  the 


96  APPROACHING   AND   RECEDING   ACTION. 

tooth.  Had  BG  been  greater,  GHmust  have  been  less,  so  that  the 
reversed  face  through  H  would  not  have  passed  through  P,  but  be- 
tween P  and  G,  and  the  case  would  have  been  impracticable,  without 
reducing  the  pitch  and  giving  both  wheels  more  teeth. 

But  if  BG  had  been  less  than  half  the  thickness  of  the  tooth,  the 
tooth  itself  could  have  been  made  higher  by  extending  the  face  above 
P,  or  it  might  be  left  of  the  height  BP,  which  would  have  given  it 
some  thickness  at  the  top,  as  in  the  next  figure. 

187.  Clearance.— The  acting  flank  is  EP ;  but  in  order  to  let  the 
teeth  of  the  other  wheel  pass,  the  hypocycloid  is  continued  to  7,  mak- 
ing the  depth  of  the  space  a  little  greater  than  PG  ;  the  difference  is 
called  clearance,  and  a  similar  provision  is  made  in  the  other  wheel 
by  cutting  in  radially,  as  shown  at  A,  H,  B,  a  little  below  the  pitch 
circle.     The  tooth  of  LM  is  completed  by  bisecting  the  pitch-arc  AE 
at  T,  and  drawing  the  curves  AK,  FJ,  similar  to  EL 

188.  A  Practical  Case. — Limiting  cases  like  the  preceding  are  to  be 
avoided  in  practice.     A  pointed  tooth  is  bad,  as  being  weak  and  liable 
to  wear  at  the  top.     And  even  if  it  be  not  pointed,  the  angle  of  action 
should  be  greater,  as  otherwise  the  least  wear  at  the  top  reduces  the 
face  below  the  requisite  height,  and  causes  one  tooth  to  quit  correct 
driving  contact  before  the  next  one  properly  begins  to  act.     We  say 
correct  driving  contact,  for  it  will  be  seen  that  if  in  Fig.  113  we  sup- 
pose all  the  teeth  to  be  removed  except  the  pair  in  contact  at  P9  the 
face  EB  would  push  the  flank  IE  out  of  its  way,  even  if  the  height 
were  slightly  reduced,  but  the  two  acting  curves  would  not  be  tangent 
to  each  other,  and  the  velocity  ratio  would  not  remain  constant,  but 
the  speed  of  the  wheel  C  would  diminish. 

A  reasonable  case  is  shown  in  Fig.  114 ;  the  arc  of  action  in  this 
instance  is  1-J-  times  the  pitch,  and  drawing  the  radial  line  PS,  we 
find  BG  to  be  much  less  than  ^  BH,  thus  enabling  us  to  give  the 
tooth  a  substantial  thickness,  PK,  at  the  top. 

189.  Approaching  and  Receding  Action. — In  Figs.  113  and  114,  the 
action  takes  place  wholly  on  one  side  of  the  line  of  centres.     If  RN 
be  the  driver  (the  directions  being  as  shown  by  the  arrows),  the  action 
begins  at  A  and  ends  at  P,  the  point  of  contact  continually  receding 
from  the  line  of  centres,  in  which  case  AB,  AE,  are  called  arcs  of  re- 
cess, or  of  receding  action.     If  LM  drive  (in  the  opposite  direction), 
the  action  begins  at  P,  ending  at  A  :  the  point  of  contact  is  always 
approaching  the  line  of  centres,  and  AB,  AE,  are  then  called  arcs  of 
approach,  or  of  approaching  action. 

It  has  been 'found  by  experience  that  the  friction  is  greater  and 
more  injurious  in  the  latter  case  than  in  the  former :  when  such 


A   PRACTICAL   CASE. 


97 


wheels  as  those  under  consideration  are  used,  therefore,  the  one  whose 
teeth  have  faces  only  should  always  drive. 

190.  But  even  then  there  is  one  drawback,  which  will  be  understood 
by  reference  to  Fig.  112.  The  longer  the  arc  of  action,  the  longer  the 
face  of  the  tooth,  and  the  greater  the  obliquity  of  the  line  of  action, 
that  is,  its  inclination  to  TT,  the  common  tangent  of  the  pitch  cir- 
cles. Now  the  pressure,  as  well  as  the  motion,  is  transmitted  in  the 
line  of  action,  and  the  greater  its  obliquity,  the  greater  will  be  the 
component  of  pressure  in  the  line  of  centres,  tending  to  cause  friction 
and  wear  in  the  bearings. 

The  amount  of  sliding  also  increases  more  rapidly  as  the  point  of 

C 


contact  recedes  from  the  line  of  centres,  so  that  upon  the  whole  such 
wheels  are  better  suited  for  use  in  light  mechanism  where  the  teeth 
can  be  made  small  and  numerous,  and  smoothness  of  action  is  impor- 
tant, than  for  the  transmission  of  heavy  pressures. 

Teeth  with  loth  Faces  and  Flanks. 

191.  It  will  now  readily  be  seen  that  by  using  another  describing 
circle  on  the  other  side  of  the  point  of  contact  of  the  pitch  circles, 
thus  giving  both  faces  and  flanks  to  the  teeth  of  each  wheel,  two 
things  will  be  accomplished  :  a  given  angle  of  action  may  be  secured 
with  shorter  faces,  and  therefore  with  less  sliding,  and  this  angle  will 
be  divided  into  an  angle  of  approach  and  an  angle  of  recess,  thus  en- 
abling us  to  use  either  wheel  as  the  driver. 


98 


TEETH  WITH  BOTH  FACES  AND  FLANKS. 


If  a  wheel  has  both  to  drive  and  to  follow,  the  arcs  of  approach  and 
of  recess  may  be  made  equal  or  nearly  so,  but  if  one  wheel  of  a  pair  is 
always  to  be  the  driver,  it  may  be  desirable  to  make  the  arc  of  recess 
the  greater,  in  order  to  reduce  the  amount  of  the  more  detrimental 
friction. 

192.  The  construction  is  shown  in  Fig.  115  ;  all  that  relates  to  the 
face  BP  for  RN,  and  to  the  flank  EP  for  LM,  is  precisely  the  same 
as  in  Fig.  113,  and  the  lettering  being  made  so  far  to  correspond  in 

N 


FIG.  115. 

the  two  diagrams,  no  further  explanation  is  needed  in  reference  to 
those  curves.  To  complete  the  teeth,  another  describing  circle  is 
used  on  the  opposite  side  of  the  pitch  circumferences,  which  generates 
the  face  OF  for  LM  and  the  flank  OK  for  RN. 

If  we  assume  the  arc  of  action  on  that  side  of  CD,  as  for  instance 
A F  or  AK,  the  possibility  of  securing  it  with  a  given  number  of 
teeth  is  at  once  ascertained  by  making  the  arc  AO  equal  to  AF,  and 
drawing  OC  to  cut  LM  in  J.  If  FI  be  less  than  half  the  thickness  of 
the  tooth  which  is  required  by  the  given  pitch,  or  equal  to  it,  the 
construction  is  possible,  the  tooth  in  the  latter  event  being  pointed  ; 
if  greater,  it  is  impracticable. 

Should  it  prove  to  be  feasible,  we  have  only  to  draw  the  epicycloid 
OF,  which  joined  to  EP  completes  the  outline  of  the  tooth  for  LM, 
and  the  hypocycloid  OK,  joining  the  latter  to  BP,  which  finishes  the 
outline  of  the  tooth  of  RN. 


COMPLETE   WHEELS — A   PRACTICAL   CASE. 


99 


That  is  to  say,  these  are  the  whole  of  the  acting  outlines  ;  the 
flanks  are,  as  already  explained,  extended  to  a  greater  depth  in  order 
to  give  clearance. 

193.  The  operation  will  be  readily  traced,  as  in  the  diagram  the 
acting  side  of  a  tooth  of  each  wheel  is  drawn  in  two  positions,  show- 
ing the  state  of  affairs  at  the  beginning  and  at  the  termination  of  the 
action  respectively.  Supposing  RN  to  drive,  the  action  begins  at  0, 
the  driver's  flank  pushing  the  face  of  the  follower,  and  the  point  of 
contact  moving  in  the  arc  OA,  until  the  points  K  and  F  meet  at  A. 
The  face  of  the  driver  then  urges  the  follower's  flank,  the  point  of 
contact  now  traveling  in  the  arc  AP,  and  at  P  the  action  ends. 


We  see,  then,  that  the  angle  of  approach  depends  upon  the  length 
of  the  follower's  face,  and  the  angle  of  recess  upon  that  of  the  driver's 
face  :  and  if  these  lengths  be  assumed  or  given,  the  angles  are  easily 
found.  For  instance,  had  the  length  of  the  follower's  face  been  as- 
signed, as  ES,  then  a  circle  described  about  C  through  S  cuts  the  de- 
scribing circle  of  that  face  in  0,  determining  A  0  the  length  of  the 
arc  of  approach,  to  which  AF  and  A  K  are  to  be  made  equal. 

194,  A  Practicable  Example. — The  diagram,  Fig.  115,  is  drawn 
without  regard  to  practical  proportions,  the  only  object  being  to  illus- 
trate the  eonstruction  clearly  ;  but  in  Fig.  116,  we  have  shown  a 
feasible  case.  The  cut  is.  half  size,  and  the  conditions  assigned  are  as 
follows  : 


100 


INTERCHANGEABLE   WHEELS. 


Distance  between  centres,,  27  inches. 
Wheels  to  have  63  and  45  teeth  respectively. 
The  smaller  wheel  to  be  the  driver. 
Whole  arc  of  action  to  be  2s  times  the  pitch. 
Angle  of  recess  to  be  ^  greater  than  angle  of  approach. 
We  have,  then, 

63  :  45  ::  7  :  5,    7  +  5  =  12,    f|  =  2}, 


also 


Again, 


2}  x  7  =  15|,  =  radius  of  larger  pitch  circle. 
2}  x  5  =  11},  =      "      "  smaller  "      " 


which  is  to  be  divided  into  parts  in  the  ratio  of  3  to  4  ; 
whence, 

3  +  4=7,     V  -v-  7  =  f ; 
and 

t  x  3  =  li  times  the  pitch  =  angle  of  approach, 


1x4=1* 


recess. 


Interchangeable  Wheels. 

195.  Inasmuch  as  the  face  and  the  flank  which  act  upon  each  other 
are  generated  by  the  same  describing  circle,  it  makes  no  difference 


FIG.  117.  FIG.  118.  FIG.  119. 

whether  the  diameter  of  the  one  which  traces  the  other  face  and  flank 
be  the  same  or  not,  in  laying  out  a  single  pair  of  wheels,  and  in  Fig. 
115,  the  describing  circles  are  of  different  diameters.  But  for  the  very 
reason  just  stated,  it  is  clear  that  if  we  wish  to  make  a  number  of 
wheels,  any  one  of  which  will  gear  with  any  other  one,  we  must  use 
the  same  describing  circle  for  all  the  faces  and  all  the  flanks. 

196.  Size  of  the  Describing  Circle. — In  making  such  a  set  of  inter- 
changeable wheels,  the  question  at  once  arises,  how  large  should  the 
describing  circle  be  ? 


LIMITING   DIAMETEE   OF   DESCEIBING   CIKCLE.  101 

The  answer  to  this  depends  upon  properties  of  the  hypocycloid, 
which  are  illustrated  in  the  next  three  diagrams.  In  Fig.  117,  the 
describing  circle  is  half  as  large  as  the  pitch  circle  ;  and  the  generated 
curve  degenerates  into  a  right  line,  so  that  the  tooth,  having  radial 
flanks,  is  comparatively  weak  at  the  root;.  In  Fig.  118,  the  describ- 
ing circle  is  smaller,  and  the  flank  curves  away  from  the  radius,  out- 
wardly with  respect  to  the  body  of  the  tooth,  as  it  recedes  from  the 
pitch  circle,  giving  a  stronger  form.  In  Fig.  119,  on  the  other  hand, 
the  describing  circle  is  larger,  and  the  flank  curves  in  the  opposite 
direction,  rendering  the  tooth  both  weak  and  difficult  to  make. 

The  safe  practical  deduction  would  seem  to  be,  that  the  dia- 
meter of  the  describing  circle  should  not  be  more  than  half  that 
of  the  pitch  circle  of  the  smallest  wheel  of  a  set.  Still,  it 
will  be  found  that  if  it  be  made  five-eighths  instead  of  one- 
half  that  diameter,  the  curvature  of  the  flanks  will  not  be  so  great, 
with  the  customary  proportions  of  height  to  thickness  of  the  teeth, 
as  to  make  the  spaces  any  wider  at  the  bottom  than  at  the  pitch  cir- 
cle ;  the  teeth  can  therefore  be  made  as  usual  by  means  of  a  milling 
cutter,  and  a  describing  circle  of  the  size  last  mentioned  has  been 
employed  with  excellent  results. 

197.  In  special  constructions,  as,  for  instance,  in  laying  out  a  single 
pair  of  wheels  for  which  cutters  are  to  be  made  expressly,  good  results 
for  general  purposes  may  be  attained  by  the  use  of  two  describing  cir- 
cles, the  diameter  of  each  being  three-eighths  that  of  the  pitch  circle 
within  which  it  rolls. 

However,  with  a  given  arc  of  action,  the  face  is  shorter,  and  the 
obliquity  of  the  line  of  action  less,  the  larger  the  describing  circle. 
Consequently  in  very  delicate  mechanism,  as,  for  example,  in  watch- 
work  or  clock-work  of  the  finest  grades,  the  advantages  thus  gained 
may  make  it  advisable  to  use  teeth  of  the  form  shown  in  Fig.  119, 
notwithstanding  the  difficulty  of  making  them,  and  their  inherent 
weakness  ;  the  latter  may  be  to  some  extent  obviated  by  using  large 
fillets  at  the  junction  of  the  sides  and  bottom  of  the  spaces,  which  is 
quite  admissible,  because,  as  already  shown,  the  acting  flank  is  com- 
paratively short,  and  the  exact  outline  of  the  clearing  space  is  of  no 
consequence,  so  long  as  the  space  is  great  enough.  « 

198.  Rack  and  Wheel. — A  rack  is  simply  an  infinitely  large  wheel. 
The  curvature  of  a  circle  diminishes  as  the  radius  increases,  and  dis- 
appears when  the  radius  becomes  infinite.     Thus  the  pitch  line  of  a 
rack  is  only  a  straight  tangent  to  the  pitch  circle  of  the  wheel  with 
which  it  works,  and  the  line  of  centres  becomes  a  perpendicular  to 
this  pitch  line,  passing  through  the  centre  of  the  wheel. 


102 


BACK  AND   WHEEL. 


The  rack  will  travel  through  a  distance  equal  to  the  circumference 
of  the  pitch  circle  of  the  wheel  during  one  revolution  of  the  latter, 

whatever  the  num- 
ber of  teeth,  and 
in  the  same  propor- 
tion for  any  frac- 
tion of  a  revolu- 
tion. The  pitch  of 
the  rack  teeth, 
therefore,  is  found 
by  rectifying  the 
pitch  arc  of  the 
wheel,  whatever 
that  may  be,  and 
setting  off  that  length  upon  the  pitch  line.  The  construction  is 
shown  in  Fig.  120 ;  the  two  describing  circles  are  here  made  of  the 
same  diameter,  and  it  is  clear  that  if  the  same  circle  be  used  to  gen- 
enite  the  faces  and  flanks  of  a  set  of  wheels,  any  one  of  them  will  gear 
with  the  rack  if  the  pitch  be  also  the  same. 

199.  Evidently,  both  faces  and  flanks  of  the  rack  teeth  are  cycloids, 
being  generated  by  the  rolling  of  a  circle  upon  the  pitch  line.     If  the 
length  of  the  face  be  assumed,  as  FFFfor  instance,  a  line  parallel  to 
RN,  through  Fthc  highest  point,  cuts  the  pitch  circle  in  P,  thus  de- 
termining AP,  to  which  AB  must  be  made  equal,  and  fixing  the  part 
of  the  action  which  will  take  place  on  the  right  of  CD.     Or  if  AB 
be  assigned,  we  make  AP  equal  to  it,  thus  ascertaining  the  necessary 
length  of  face.     In  either  case,  PS  is  now  to  be  drawn  perpendicular 
to  the  pitch  line,  which  it  cuts  at  G  ;  and  as  in  the  preceding  con- 
structions, BG  cannot  be  greater,  and  should  be  less,  than  half  the 
thickness  of  the  tooth  as  determined  by  the  pitch.     The  part  of  the 
action  which  will  take  place  on  the  left  of  CD,  depends  upon  the 
length  of  the  face  of  the  wheel  tooth,  and  is  ascertained  as  in  the 
cases  previously  explained. 

200.  If  in  constructing  the  teeth  for  a  pair  of  wheels,  we  employ 
two  describing  circles,  the  diameter  of  each  being  one-half  that  of  the 
pifcch  circle  within  which  it  rolls,  the  teeth  of  each  wheel  will  have 
radial  flanks.     A  similar  course  may  be  pursued  in  laying  out  a  rack 
and  wheel,  as  shown  in  Fig.  121.     The  describing  circle  whose  diam- 
eter is  A  C,  generates  the  radial  flanks  of  the  wheel-teeth  and  the 
cycloidal  faces  of  the  rack-teeth. 

The  diameter  of  the  pitch-circle  of  the  rack  being  infinite,  one-half 
of  it  is  also  infinite,  and  EN  is  therefore  the  describing  line  for  the 


ARBITRARY   PRACTICAL   PROPORTIONS. 


103 


faces  of  the  wheel-teeth,  which  consequently  aro  involutes  of  the  pitch- 
circle  LM,  and  for  the  flanks  of  the  rack-teeth.     The  latter  being 


M 


FIG.  121. 

radii  of  the  infinite  circle  RN,  would,  strictly,  be  simply  straight 
lines  perpendicular  to  it,  as  shown  in  the  teeth  on  the  right  of  CD ; 
and  they  were  formerly  made  so.  But  when  these  parts  of  the  teeth 
are  in  action,  the  point  of  contact  moves  in  the  describing  line,  that 
is  to  say,  in  the  pitch  line  RN  itself. 

The  fact  is,  that  the  acting  flank  of  the  rack-tooth  degenerates  into 
a  point ;  thus  a  marking  point  carried  by  RN,  in  moving  from  A  to 
0,  traces  on  the  plane  of  the  wheel,  which  turns  as  indicated  by  the 
arrow,  the  involute  F09  the  arc  A  F  being  equal  to  AO  :  while,  hav- 
ing no  motion  relatively  to  the  plane  of  the  rack,  it  marks  on  that 
plane  simply  the  point  0.  The  action  is  consequently  bad,  this  point 
being  subjected  to  excessive  wear  :  but  the  flank  of 
the  rack  tooth  may  be  made,  as  shown  on  the  left 
of  OD,  an  arc  of  a  circle  whose  centre  is  on  RN, 
and  radius  equal  to  the  radius  of  curvature  of  the 
face  of  the  wheel-tooth  at  its  highest  point.  This 
radius  is  found  thus  :  let  V  be  the  highest  point  of 
the  involute  tooth ;  draw  through  V  a  tangent  to 
the  pitch-circle  LM,  lying,  as  shown,  on  the  concave 
side  of  the  involute,  and  find  the  point  of  tangency 
T  \  then  VTis  the  required  radius  of  curvature. 

201,  Arbitrary  Proportions. — It  is  not  necessary 
in  all  cases,  to  pay  particular  attention  to  the  rela- 
tive amounts  of  the  approaching  and  the  receding  action.  And  it  is 
a  very  common  practice  to  make  the  whole  radial  height  of  the  tooth 
a  certain  fraction  of  the  pitch ;  the  part  without  the  pitch  circle 
being  a  little  less  than  that  within,  by  which  the  clearance  is  pro- 
vided for.  In  Fig.  122,  these  parts  are  marked  h  and  d  respectively. 
I  being  the  whole  height.  Three  of  these  arbitrary  proportions,  as 


FIG.  122. 


104  BACKLASH. 

they  may  be  called,  which  have  been  extensively  adopted,  are  as  fol- 
lows : 

1.  I  =  -f  pitch  ;  h  :  d  :  :  11  : 13.     (b  =  TV  pitch.) 

2.  1=  f  pitch;  hid::    4:    5.     (b  =  &  pitch.) 

3.  I  =  j\  pitch  ;  h  :  d  :  :   3  :    4.     (b  =  T1T  pitch.) 

The  whole  angle  of  action,  as  well  as  the  ratio  of  the  approach  to 
the  recess,  will  of  course  vary  according  to  the  numbers  of  the  teeth, 
in  the  use  of  any  such  system  :  but  either  of  these  rules  will  give  sat- 
isfactory results  for  most  purposes,  the  wheels  acting  as  drivers  or 
followers  indifferently,  provided  that  there  are  at  least  twelve  teeth 
upon  the  smallest  wheel.  Less  than  that  should  not  be  used  unless 
it  is  necessary  :  sometimes,  however,  the  use  of  lower  numbers  cannot 
be  avoided,  in  which  event  it  will  often  be  requisite  to  extend  the 
faces  of  the  pinion's  teeth  beyond  these  limits ;  and  the  proper  length 
should  be  determined  as  before  explained. 

202.  It  was  also  formerly  the  custom  to  make  the  backlash  a  defi- 
nite fraction  of  the  pitch,  which  we  have  added  above  in  parenthesis 
as  usually  given  in  connection  with  each  of  the  preceding  rules.     But 
although  these  values  may  have  been  proper  in  many  cases,  as  allow- 
ing for  imperfections  of  workmanship  in  wooden  mill-work  or  when 
the  wheels  were  simply  to  be  cast,  it  is  certain  that  they  are  in  many 
cases  too  large,  if  the  teeth  are  to  be  cut  with  the  slightest  pretension 
to  accuracy.     Nor  does  there  appear  to  be  any  reason  why  the  back- 
lash should  vary  directly  with  the  pitch  ;  on  the  contrary,  it  seems 
almost  self-evident  that  the  coarser  the  pitch,  the  smaller  will  be  the 
proportion  borne  to  it  by  any  unavoidable  error.     From  this  point  of 
view,  it  appears  more  reasonable  to  say  that  the  backlash  should  vary 
inversely  as  the  pitch  ;  and  perfectly  safe  to  insist  that  it  ought  in 
every  case  to  be  as  small  as  the  skill  of  the  workman  will  enable  him 
to  make  it  with  the  facilities  at  command. 

Since  theoretically  the  teeth  may  be  in  contact  on  both  sides  at  once, 
we  have  in  the  diagrams  entirely  disregarded  the  backlash.  If  it  were 
introduced,  the  constructions,  evidently,  would  be  modified  only  in 
this  respect,  that  the  "  thickness  of  the  tooth  as  required  by  the  pitch," 
instead  of  being  exactly  half  the  pitch  arc,  would  be  less  than  that  by 
just  the  amount  of  backlash  allowed. 

3.  Annular  Wheels. 

203.  In  Fig.  123,  the  smaller  pitch  circle  lies  within  the  greater, 
which  it  touches  internally  ;  the  teeth  of  the  outer  wheel  being  there- 
fore formed  on  the  concave  circumference  of  an  annular  rim.     But 
neither  the  mode  of  generating  the  tooth-outlines,  nor  the  nature  of 


ANNULAR  WHEELS. 

their  action,  are  in  any  way  changed;  two  describing  circles  are 
shown,  each  of  which  generates  a  face  for  one  pitch  circle  and  a  flank 
for  the  other, — and  in  short,  a  comparison  of  this  diagram  with  Fig. 
115,  which  is  lettered  similarly  throughout,  will  show  that  the  two 
are  identical  in  all  particulars  relating  to  the  construction. 

It  will  also  be  ob- 
served that  the  contour 
of  the  annular  wheel, 
in  respect  to  the  forms 
of  the  acting  curves,  is 
identical  with  that  of 
an  ordinary  spur  wheel 
having  the  same  pitch 
and  describing  circles, 
the  tooth  of  the  one  cor- 
responding precisely  to 
the  space  of  the  other.  FIG.  123. 

The  describing  circles  in  the  figure  are  of  different  diameters  :  but 
they  might  have  been  equal,  and  it  will  readily  be  seen  that  the  spur 
wheels  which  are  thus  made  interchangeable  with  each  other  (195),  may 
also  be  made  interchangeable  with  annular  ones.  But  in  the  construc- 
tion of  inside  gearing  the  diameters  of  the  describing  circles  are  limited 
by  considerations  which  relate  entirely  to  the  peculiar  condition  of 
internal  tangency  between  the  pitch  circles.  The  manner  in  which 
these  limits  are  determined,  will  be  most  clearly  seen  by  first  regard- 
ing the  tooth-outlines  as  generated 
in  another  way. 

204.  Intermediate  Describing  Cir- 
cle.—In  Fig.  124,  let  C  be  the 
centre  of  the  inner  pitch  circle,  D 
that  of  the  outer,  and  A  OL  a  curve 
lying  between  the  two  circumfer- 
ences, tangent  to  both  at  A,  and 
capable  of  rolling  in  contact  with 
them.  Let  the  arcs  AK,  AF9  of 
the  pitch  circles,  be  equal  to  each 
other,  and  during  the  rotation  indi- 
cated by  the  arrow,  let  a  marking 
point  be  carried  from  A  to  0,  by  the 
describing  curve  :  it  will  in  its 
progress,  evidently,  trace  upon 
the  planes  of  rotation,  the  face  OK  for  the  pinion,  and  the  face  OF  for 


FIG.  124. 


106  INTERMEDIATE   DESCRIBING   CIRCLE. 

the  annular  wheel.  If  these  be  used  as  the  outlines  of  teeth,  it 
follows  from  the  mode  of  their  generation  that  they  will  transmit  ro- 
tation with  a  constant  velocity  ratio,  whatever  the  form  of  the  de- 
scribing curve  or  of  the  locus  of  contact :  if  the  pinion  drive,  the 
action  begins  at  A,  and  ends  at  0. 

Now,  if  it  be  required  that  these  faces  shall  be  epicycloidal,  the 
curve  AOL  must  be,  as  in  the  figure,  a  circle  whose  centre-./?  lies 
between  C  and  D.  It  is  true  that  OK  and  OF  may  then  be  generated 
in  a  different  manner  ;  which,  however,  though  an  important  coinci- 
dence, does  not  alter  the  fact  that  the  driving  contact  between  these 
two  curves  is  due  solely  to  the  mode  of  generation  here  explained. 
And  it  will  subsequently  be  seen  that  annular  wheels  may  be  required 
to  work  under  conditions  which  can  be  satisfied  only  by  this  method 
of  construction  :  upon  which  also  depends  the  determination  of  the 
limiting  diameters  of  the  describing  circles  in  either  method. 

205.  Limiting  Diameter  of  Intermediate  Describing  Circle. — In  making 
these  determinations,   we  avail   ourselves  of  the  peculiar  property 
above  alluded  to,  viz  :  that  every  epicycloid,  internal  or  external,  as 
well  as  every  hypocycloid,  is  capable  of  two  generations.     (See  Ap- 
pendix,) 

The  face  OF  being  now  a  true  hypocycloid,  may  be  generated  not 
only  by  the  intermediate  circle  whose  centre  is  E,  but  by  another 
whose  radius  is  ED.  In  Fig.  124,  AE  is  equal  to  CD,  whence  ED  = 
A  C  the  radius  of  the  smaller  pitch  circle.  Consequently  on  rolling 
the  pinion  within  the  wheel,  the  hypocycloidal  path  traced  by  the 
point  A  of  the  former,  coincides  with  the  face  AB  of  the  adjacent 
tooth  of  the  wheel.  The  radius  AE  may  be  increased,  in  which  case 
this  path  AM  will  lie  to  the  right  of  AB  ;  but  ;f  it  be  decreased, 
AM  will  lie  to  the  left  of  AB,  that  is,  within  the  body  of  the  wheel's 
tooth,  which  is  clearly  impracticable.  In  this  construction  then  we 
have  the  limit,  that 

The  radius  of  the  intermediate  describing  circle  can  not  he  less  than 
the  line  of  centres. 

206.  Drawing  through  0  a  circle  about  centre  D,  it  cuts  the  inner 
pitch  circle  in  R ;  at  which  point  the  action  begins,  the  face  KO 
then  having  the  position  RN. 

A  marking  point  at  R,  carried  by  the  inner  pitch  circle  as  a  de- 
scribing curve,  will,  in  going  to  A,  obviously  generate  the  face  R8 
similar  to  OF;  against  Avhich  this  point  R  of  the  pinion  will  act 
during  the  approach,  the  locus  of  contact  being  the  arc  RA. 

The  face  of  the  pinion  having  now  reached  the  position  AP,  the 
receding  action  begins  ;  and  here  a  phenomenon  peculiar  to  inside 


TEETH   WITH   DOUBLE   CONTACT. 


107 


gearing  presents  itself.  For  the  face  AP  is  not  only  an  internal 
epicycloid  whose  generating  circle  is  A  OL,  but  an  external  one  which 
may  be  generated  by  a  circle  whose  radius  is  AH,  equal  to  EC.  It 
will,  therefore,  work  correctly  with  a  flank  A  Q  traced  by  tho  same 
describing  circle.  This  action  ter- 
minates at  T,  the  point  in  which 
a  circle*  through  0  about  C  cuts  the 
circumference  of  the  exterior  describ- 
ing circle,  and  during  its  continuance 
the  locus  of  contact  is  the  arc 
AT. 

This  flank  may  or  may  not  be 
used ;  but  if  it  be,  since  the  action 
between  the  faces  AP,  AB,  pre- 
viously explained,  also  begins  at  A, 
the  singular  fact  appears,  that  while 
the  pinion  is  turning  through  the 
angle  PCT,  its  face  has  two  points 
of  driving  contact.  This  circum- 
stance is  of  some  practical  impor- 
tance, not  only  on  account  of  tho  FIG.  125. 
division  of  the  pressure,  but  also  as  affecting  tho  resultant  obliquity 
of  the  line  of  action. 

207.  Limiting  Diameters  of  Exterior  and  Interior  Describing  Circles. 
— In  Fig.  125,  the  radius  AE  of  the  intermediate  describing  circle  is 
greater  than  CD.     The  face  OF  may  also  be  generated  by  an  interior 
describing  circle  whose  radius  A  G  is  equal  to  ED,  and  OJTby  an 
exterior  one  of  which  the  radius  is  AH,  equal  to  EC.     In  all  that 
relates  to  the  action  the  only  new  feature  is  that  the  pinion  now  has 
a  flank  of  sensible  magnitude,  traced  by  the  circle  whose  centre  is  G  ; 
as  will  readily  be  seen,  the  lettering  throughout  being  similar  to  that 
of  Fig.   124.     Now,  either  radius,  AG  or  AH,  may  be  diminished 
without  changing  the  other  ;  the  only  result  being  that  the  faces  OF, 
OK,  as  shown  in  dotted  lines,  will  not  be  in  contact  during  the  recess, 
and  will  consequently  act  only  against  the  corresponding  flanks,  as  in 
Fig.  123.     But  if  either  of  these  radii  be  increased  without  diminish- 
ing the  other  to  the  same  extent,  it  is  apparent  that  the  faces  OF, 
OK,  will  intersect  each  other,  rendering  the  construction  impractica- 
ble :  and  we  have  as  a  limit,  that 

TJie  sum  of  the  radii  of  the  two  describing  circles  cannot  ~be  greater 
than  the  line  of  centres. 

208.  This  holds  true  when  either  radius  vanishes,  the  other  then 


108 


LIMITING   DIAMETERS   OF   DESCRIBING   CIECLES. 


becoming  equal  to  the  line  of  centres.     Thus,  if  in  Fig.  125,  we  make 
AE  =  AD,  we  shall  have 

ED  =  0  =  AH,     whence    AG  =  EG  =  CD. 
If,  on  the  other  hand,  we  make  AE  =  AC,  we  shall  have 

EG  =  0  =  AG,     whence    A  H  =  ED  —  CD. 
These  two  limiting  cases  are  illustrated  in  Figs.  126  and  127  respect- 
ively.    In  the  former,  it  is  apparent  that  when  the  pinion  drives  the 
arc  of  recess  is  greater  than  the  arc  of  approach,  but  since  the  action 


FIG.  126. 


FIG.  127. 


during  the  recess  is  confined  to  the  single  point  0  of  the  pinion's  teeth, 
the  advantage  thus  gained  is  more  than  neutralized.  The  fact  that  the 
pinion  can  have  no  face,  will  be  seen  from  the  consideration  that  if  there 
were  one,  it  must  be  tangent  at  0  to  the  radius  OC,  and  would,  there- 
fore lie  within  the  body  of  the  adjacent  tooth  of  the  annular  wheel. 
In  Fig.  127,  on  the  other  hand,  the  wheel  can  have  no  face,  for  a 

similar  reason.  The  pinion 
consequently  having  no 
flank,  there  is  no  approach- 
ing action,  but  its  face  has 
two  points  of  driving  con- 
tact during  a  part  of  the 
recess. 

209.  It  is  evidently  more 
frequently  practicable  in 
inside  than  in  outside  gear- 
ing, to  secure  an  angle  of 
recess  greater  than  the 
pitch,  and  thus  to  avoid 
altogether  the  more  injurious  friction  of  approaching  action.  An  in- 
stance of  this  is  shown  in  Fig.  128,  where,  the  wheel  driving,  an  in- 


FIG.  128. 


SPECIAL   CASES. 


109 


terior  describing  circle  only  is  employed,  whose  radius  is  in  this  case 
half  that  of  the  inner  pitch  circle,  thus  giving  the  pinion  radial 
flanks.  •  Attention  is  called  to  the  fact  that  when  the  tooth  thus  has 
no  face,  it  should,  nevertheless,  be  allowed  to  project  beyond  the  pitch 
circle  for  the  sake  of 
strength,  the  corner  being 
finished  by  a  circular  arc 
tangent  to  the  radius  at  its 
extremity. 

Fig.  129  shows  the  ap- 
pearance of  an  annular 
wheel  and  pinion  which 
differ  but  little  in  size  ;  the 
teeth  are  necessarily  very 
short  in  order  that  they 
may  escape  from  engage- 
ments and  pass  each  other. 
Whether  they  will  do  so  or 
not,  if  the  height  be  as-  FIG.  129. 

sumed,  is  readily  determined  by  constructing  the  epitrochoid  traced 
by  the  highest  point  of  either  tooth  when  its  pitch  circle  is  rolled 
upon  the  other  one  ;  this  path,  obviously,  must  not  intersect  the  tooth 
outline  of  the  engaging  wheel :  and  the  clearing  spaces  of  both  the 
wheel  and  the  pinion  must  also  be  such  as  not  to  touch  the  epitro- 
choids  thus  described. 


CHAPTER  VII. 


LOW-NUMBERED    PINIONS— PRACTICAL    LIMIT    OF     OBLIQUITY  —  PIN- 
IONS WITH  3,  4,  and  5  LEAVES — LEAST  NUMBER  THAT  CAN  BE 

USED  WITH  A  GIVEN  DRIVER — THE  TWO-LEAVED'  PINION — LOW 
NUMBERS  IN  INSIDE  GEAR  —  VARIOUS  EXAMPLES  —  INFERIOR 
AND  SUPERIOR  LIMITS — TWO-LEAVED  PINIONS  IN  INSIDE  GEAR — 
LEAST  NUMBER  THAT  CAN  BE  USED  WITH  A  GIVEN  FOLLOWER. 


On  the  Use  of  Low-numbered  Pinions. 

210.  In  the  operation  of  epicycloidal  teeth,  the  obliquity  of  the  line 
of  action  is  continually  varying  ;  diminishing  during  the  approach,  it 
becomes  zero  when  the  point  of  contact  reaches  the  line  of  centres,  and 
again  increases  during  the  recess. 

If  wheels  are  to  do  heavy  work,  it  has  been  found  by  experience  that 
the  mean  obliquity  should  not  in  general  exceed  about  15°,  nor  the 
maximum  about  30°.  A  high  maximum  is  less  objectionable  when 
several  pairs  of  teeth  are  engaged  at  once,  since  the  greatest  portion  of 
the  pressure  will  be  acting  less  obliquely.  Such  distribution  of  the 
pressure,  it  will  readily  be  seen,  is  more  often  to  be  attained  in  inside 
gearing,  and  the  obliquity  is  one  serious  disadvantage  when  low-num- 
bered pinions  act  in  outside  gear.  Another  is  the  excessive  amount 
of  sliding,  due  to  the  necessarily  great  length  of  the  faces  of  the  teeth. 
And  from  both  combined  is  deduced  the  practical  rule  that,  in  mill- 
work  and  machinery  in  general,  no  pinion  of  less  than  twelve  teeth 
should  be  used  if  it  be  possible  to  avoid  it. 

But  it  is  not  always  possible  ;  and  in  lighter  mechanism,  such  as 
clock-work,  it  is  often  necessary  to  use  much  lower  numbers.  In  work 
of  this  description  a  greater  obliquity  is  often  admissible  ;  and  when 
it  is  not  so  considered,  the  convexity  of  the  flanks  caused  by  using  a 
large  interior  describing  circle,  is  not  so  objectionable  when  the  work 
to  be  done  is  light,  and  strength  of  form  not  an  imperative  necessity. 

For  the  present,  then,  we  will  confine  ourselves  to  the  following 
limits,  viz  :  that  the  maximum  obliquity  shall  not  exceed  36°,  and 


LOW-NUMBERED   PINIONS. 


Ill 


that  the  diameter  of  the  describing  circle  for  the  flanks  shall  not 

exceed  -f-  that  of  the  pitch 

circle  within  which  it  rolls. 
211.  In  Fig.  130  are  shown 

two  equal  pitch  circles,  and 

two  equal  describing  circles 

of  half  the  diameter,   TAS 

being  the  common  tangent. 

Draw  the  line  of  action  with 

an  obliquity  of  30°  ;  it  cuts 

the  describing   circles  in   0 

and  P,  making  the  arcs  A  0, 

AP,  each  equal  to  60°.  Draw- 
ing <7PJ?and  DOK,  the  arcs 

AE,  AK,  are  each  equal  to 

30°,  and  we  thus  have  two 

similar  wheels  which  will  just 

work,  the  angles  of  approach 

and  of  recess  being  each  equal  FIG.  130. 

to  half  the  pitch,  and  the  mean  obliquity  exactly  15°.     It  is  seen  to 

be  possible,  with  only  a  slight  increase  in  the  obliquity,  to  lengthen 

the  faces  and  secure  an  arc  of 
action  greater  than  the  pitch, 
as  in  practice  it  should  be. 

In  this  case  the  flanks  are 
radial.  But  if  without  chang- 
ing the  describing  circles,  we 
reduce  the  pitch  circles  to  f 
their  present  diameter,  the  arc 
AP  of  60°  will  be  equal  to  an 
arc  of  36°  on  the  new  pitch  cir- 
cumference.  It  is,  then,  prac- 
ticable  to  make  two  pinions  of 
five  teeth  each,  which,  like  those 
in  the  figure,  will  just  work,  with 
the  same  obliquity.  The  flanks 


- 


will  now  be  convex;  still,  the  di- 
ameter of  the  describing  circle  is 
within  the  assumed  limit,  being 
FIG.  LSI.  but  -jfcj-  that  of  the  pitch  circle. 

212.  In  Fig.  131  the  diameter  of  the  larger  pitch  circle  is  1}  times 
that  of  the  smaller  one  ;  also  the  diameter  of  the  upper  describing 


LOW-ls  TJMBEKED  PINIONS. 

circle  is  equal  to  CA,  so  tjiat  as  in  the  preceding  figure  the  arc  AP 
of  60°  is  equal  to  the  arc  AE  of  30°,  and  PAS,  the  maximum  obli- 
quity on  the  right  of  CD,  is  also  30°.  The  diameter  of  the  lower  de- 
scribing circle  is  -|  that  of  its  pitch  circle,  therefore  the  arc  A  0  of  73° 
is  equal  to  the  arc  AK  of  45°,  and  OAT,  the  maximum  obliquity  on 
the  left  of  CD,  is  36°.  Completing  the  construction,  we  perceive  that 
the  pinion  of  four  teeth  will  just  work  with  the  other  of  six.  The 
arcs  of  approach  and  recess  are  equal,  the  mean  obliquity  being  18° 
during  the  one  and  15°  during  the  other,  giving  an  average  of  16|-° 
for  the  whole  action  if  the  Tatter  be  made  just  equal  to  the  pitch  ;  and 
the  faces  may  also  be  made  a  little  longer  than  here  shown,  as  in  the 
previous  case. 

The  leaves  of  the  larger  pinion  still  have  radial  flanks,  but  proceed- 
ing as  before,  we  may  reduce  the  upper  pitch  circle  to  f  of  its  present 
diameter,  without  changing  the  circle  which  rolls  within  it.  TVe  shall 
then  have  the  arc  AE  equal  to  36°,  and  the  five-leaved  pinion  thus 
constructed  will  work  with  the  four- leaved  one,  the  obliquity  remain- 
ing unchanged,  since  the  points,  0,  A,  P,  retain  the  same  positions 
as  in  the  figure. 

213,  Now  produce   OA  to  I,  making  AI  =   AO.      Then,  were 
another  describing  circle  drawn  above  T8,  equal  in  diameter  to  the 
one  below  it,  the  point  /would  lie  upon  its  circumference  ;  and  since 
the  arc  AI  would  be  equal  to  the  arc  AB,  the  epicycloid  traced  by  roll- 
ing it  upon  the  lower  pitch  circle  would  extend  to  the  point  B. 

Draw  ID  ;  it  will  then  be  apparent  that  were  the  angle  IDB  equal 
to  or  less  than  the  angle  IDA,  two  four-leaved  pinions  would  be  capa- 
ble of  working  together,  all  the  acting  curves  being  traced  by  a  de- 
scribing circle  whose  diameter  is  •§  that  of  the  pitch  circles.  Now  in 
the  triangle  ADI,  the  sides  AI  and  AD,  also  their  included  angle, 
are  known  ;  whence  it  will  be  found  that  the  angle  IDA  is  22°  27' 
42",  whereas  it  should  be  22°  30'  in  order  that  one  pinion  should 
drive  the  other  through  a  total  arc  of  action  just  equal  to  the  pitch, 
even  if  the  teeth  were  pointed. 

Five,  then,  is  the  least  number  of  teeth  that  can  be  used  if  the  pin- 
ions are  to  be  alike,  and  four  will  work  with  five  or  any  higher  number. 

214.  The  force  of  the  objections  urged  against  the  use  of  small 
pinions  for  heavy  work,  is  clearly  shown  in  Fig.  130.     Letting  OA 
represent  the  pressure  when  0  is  the  point  of  contact,  this  may  be 
resolved  into  the  two  components  Om,  On ;  of  which  the  latter  tends 
simply  to  force  the  journals  apart,  thus  increasing  the  friction  in  the 
bearings.     Since  only  one  pair  of  teeth  is  in  action,  and  the  angle 
OA  T  is  30°,  this  objectionable  component  is  equal  to  half  the  total 


THREE-LEAVE  D    PIKIOH — OUTSIDE    GEAK. 


113 


pressure  at  0,  which  again  is  greater  than  that  at  the  point  K  on  the 
pitch  circumference,  in  the  ratio  of  DKto  DO. 

215.  Thus  far,  the  angles  of  approach  and  recess  have  been  made 
equal.     Should  it  be  desired  to  make  them  otherwise,  the  question 
whether  any  assumed  conditions  can  be  satisfied  or  not,  is  readily  set- 
tled by  the  construction  of  a  diagram,  as  explained  in  connection 
with  Fig.  115. 

If,  as  usual  in  such  cases,  the  angle  of  recess  is  to  be  the  greater, 
it  will  be  apparent  that  with  a  wheel  of  a  given  number  of  teeth  a 
smaller  pinion  may  in  general  be  used  to  drive  than  to  follow.  For 
the  angle  of  recess  depends  upon  the  length  of  the  face  of  the  driver's 
tooth  ;  and  the  smaller  the  pinion  the  smaller  will  be  the  describing 
circle  which  can  be  used  within  it,  and,  consequently,  the  less  will  be 
the  greatest  possible  length  of  the  face  of  the  wheel-tooth,  ^ince  the 
pitch  is  already  assigned  by  the  conditions. 

216,  An  illustration  of  this  is  incidentally  afforded  by  Fig.  132,  in 
the  construction  of  the  three-leaved  pinion.     Taking  for  its  flanks  a 
describing  circle  whose  diame- 
ter is  |  that  of  its  pitch  circle, 

and  making  OAT,  the  maxi- 
mum obliquity  on  the  left  of 
CD,  36°  as  before,  the  angle  of 
action,  KDA,  on  that  side  of  the 
line  of  centres  is  45°.  The 
pitch  arc  KB  is  120°,  there- 
fore the  angle  of  action  ADB, 
on  the  right  of  CD,  must  be 
75°.  It  will  be  found  that  this 
can  be  secured,  with  a  very  lit- 
tle to  spare,  by  using  for  its 
faces  a  describing  circle  five 
times  as  large  as  the  one  used 
for  its  flanks. 

And  keeping  to  the  limits 
heretofore  observed,  it  follows 
that  the  second  pitch  circle  can- 
not be  less  than  five  times  as  large 
as  that  of  the  pinion  ;  which, 
therefore,  will  just  work  with  a 
wheel  of  fifteen  teeth1,  and  no 

le$s,   under  the  conditions  as-  FIG.  132. 

signed.     In  other  words,  a  three-leaved  pinion  will  drive  a  wheel  of 
8 


114 


LEAST  NUMBER   OF  TEETH. 


fifteen  teeth,  the  angle  of  approach  being  |,  and  that  of  recess  f,  of 
the  pitch  angle.  Seasonably  well,  too,  in  respect  to  the  obliquity  of 
the  line  of  action  ;  during  the  approach  the  maximum  is  36°  and  the 
mean  is  18°,  during  the  recess  the  maximum  is  13°  and  the  mean  6°  : 
but  the  latter  angle  is  larger  than  the  former  in  the  ratio  of  5  to  3, 
whence  we  have,  as  the  average  obliquity  during  the  whole  action, 

18°  x  3  +  G°  x  5       84° 


3  +  5  8 

217.  It  will  be  perceived  that  the  above  construction  involves  the 
solution  of  the  following  problem,  viz.  : 

Given  the  pitch  circle,  number  of  teeth,  and  arc  of  recess,  of  the 
driver,  to  find  the  least  number  of  teeth  which  can  be  assigned  to  the 
follower.. 

This  may  be  determined  graphically,  as  shown  in  Fig.  133,  where 
CD  is  the  line  of  centres,  0  the  centre  of  the  driver,  and  AF  the 

given  arc  of  recess.  Make  FL  equal 
to  J  the  pitch,  that  is,  half  the  thick- 
ness of  the  tooth,  and  draw  through 
L  a  radial  line  of  indefinite  length  ; 
then  if  the  tooth  be  pointed,  its  point 
will  lie  upon  the  prolongation  of  CL. 
Make  A  G,  perpendicular  to  CD,  equal 
to  the  arc  AF  (see  Appendix  (1)),  and 
set  off  A  M  =  i  A  G.  With  centre  M 
and  radius  MG  —  j  AG,  describe  an 
arc  cutting  CL  produced  in  0.  Draw 
OA,  bisect  it  in  N  by  a  perpendicu- 
lar cutting  CD  in  E\  then  (see  Ap- 
pendix (1)),  the  arc  OA,  whose  centre 
is  E,  will  be  equal  to  A  G  and  therefore 
to  AF. 

Evidently,  then,  E  is  the  centre  of  a 
describing  circle  which  by  rolling  on 
the  given  pitch  circle  will  trace  an 
epicycloid  from  0  to  F,  and  this  will 
FIG.  133.  be  the  face  of  the  driver's  tooth.     Ob- 

serving the  limits  before  assigned,  the  radius  of  the  required  pitch 
circle  cannot  be  less  than  f  of  AE  ;  but  it  must  be  such  that  the  given 
pitch,  viz.,  four  times  the  arc  FL,  shall  be  an  aliquot  part  of  the  cir- 
cumference. 

It  may  be  noted,  that  the   conditions  assigned  as  above  furnish 


FIVE-LEAVED    DRIVEN 


115 


sufficient  data  to  enable  us  to  verify  the  results  by  trigonometrical 
computation,  should  it  be  considered  necessary. 

218.  By  the  above  process  we  ascertain  at  once  the  size  of  the  de- 
scribing circle  which  will  give  the  driver  a  pointed  tooth,  and  from 
this  derive  the  diameter  of  the  follower's  pitch  circle.     Should  the 
latter  correspond  to  a  fractional  number  of  teeth,  the  next  higher 
integer  must  be  used,  the  pitch  circle  being  increased  accordingly. 
In  this  case  the  diameter  of  the  describing  circle  may  be  increased  or 
not,  at  pleasure  ;  though  it  is  better  that  it  should  be,  since  then  the 
driver's  tooth  may  be  topped  off. 

And  it  may  be  necessary  to  increase  it  for  another  reason  ;  because 
the  very  process  of  determining  the  minimum  radius  AE,  Fig.  131, 
fixes  also  the  maximum  obliquity  OA  G  corresponding  thereto.  At- 
tention must,  therefore,  be  given  to  this,  for  it  is  possible  that  the 
obliquity  thus  fixed  may  be  too  great,  although  the  numbers  of  the 
teeth  be  practicable ;  and  in  that  event  the  describing  circle,  and,  if 
necessary,  the  pitch  circle  as  well,  must  be  increased  until  the  ob- 
liquity is  reduced  to  the  desired  limit. 

219.  Fig.  134  illustrates  the  converse  case  to  that  of  Fig.  132,  the 
conditions  being  that  the  wheel  of  15  teeth  is  to  drive,  and  that  the 
angle  of  recess  shall  be  f  of  the  pitch 

angle.  Making  the  construction  as 
explained  in  (217),  the  least  pitch 
circle  of  the  follower  corresponds, 
as  found  by  computation,  to  4.45 
teeth,  with  a  maximum  obliquity  of 
40°  30'  2".  Five,  therefore,  is  the 
least  practicable  number  of  leaves 
that  the  pinion  can  have,  thus  mak- 
ing the  pitch  72°,  and  the  arc  of 
recess  45°,  and  by  using  a  describing 
circle  for  their  flanks,  whose  diame- 
ter is  f  that  of  the  pitch  circle,  the 
maximum  obliquity  during  the  re- 
ceding action  is  reduced  to  36°, 
while  at  the  same  time  the  driver's 
tooth  is  made  of  reasonable  breadth  FlG- 134; 

at  the  top.  In  this  diagram,  as  in  Fig.  132,  the  angle  of  approach  is 
|  of  the  pitch  angle,  but  it  is  obvious  that  although  the  driver's  flanks 
are  radial,  the  angle  of  approach  might  be  very  considerably  increased 
in  this  case,  although  in  that  of  the  3-leaved  driver  it  could  not  be. 

220.  The  two-leaved  Pinion. — When  a  spur-wheel  is  made  in  the  or- 


116 


TWO-LEAVED   DKIVING 


dinary  way,  that  is,  with  the  teeth  in  the  same  plane,  a  pinion  of  three 
leaves  is  the  smallest  that  can  be  used  either  to  drive  or  to  follow. 
But  if  the  alternate  teeth  are  placed  in  different  planes,  a  two-leaved 
pinion  can  be  made  to  drive  in  a  very  satisfactory  manner  ;  the  ar- 
rangement is  clearly  shown  in  Fig.  135,  the  pinion  being  composed  of 
two  heart-shaped  cams  or  teeth,  U  and  W,  fixed  side  by  side  on  the 
same  shaft,  and  the  wheel  of  the  two  star-shaped  plates  similarly  fixed 
upon  another  shaft,  of  which  the  one,  M,  works  with  the  cam  U,  and 
the  other,  N,  with  W.  The  diameter  of  the  describing  circle  is  equal 
to  the  radius  of  the  pitch  circle  of  the  follower,  whose  flank  is  there- 
fore radial.  The  diameters  of  the  pitch  circles  arc  in  this  case  in  the 

ratio  of  6  to  1  ;  the  arc  AP  of  60°, 
is  therefore  equal  to  the  arc  AE  of 
30°,  and  to  the  semi-circumference 
AB.  In  the  position  here  shown 
P  is  a  point  of  contact,  but  the  epi- 
cycloid BP  is  continued  to  R,  so 
that  the  action  is  not  yet  ended, 
although  the  tooth  A  IS  is  just  be- 
ginning to  drive  a  flank  of  the  other 
plate  JVof  the  wheel. 

It  is  obvious  that  a  half  revolution 
of  the  pinion  will  bring  the  point  8 
to  the  position  R,  turning  the  wheel 
through  the  angle  ACE  of  30°  ;  G 
and  B  then  meeting  at  A,  the  ac- 
tion will  be  continuous. 

The  obliquity,  it  will  be  observed, 
is  not  great,  but  the  amount  of  sliding  is  excessive,  owing  to  the  great 
length  of  the  face  PB  as  compared  with  the  flank  PE ;  still  the  ac- 
tion is  smooth  and  noiseless,  since  the  driving  contact  is  wholly  re- 
ceding, there  being  no  arc  of  approach. 

The  number  of  teeth  in  the  wheel  may  be  reduced,  for  the  flank  need 
not  be  radial  :  by,.using  a  describing  circle  of  |  the  diameter  of  the 
pitch  circle,  and  making  AP  —  72°,  the  angle  ACE  will  become  45°, 
the  velocity  ratio  thus  obtained  being  that  of  4  to  1,  and  the  maxi- 
mum obliquity  very  little  over  36°. 

221,  Low-numbered  Pinions  in  Inside  Gear. — In  making  investiga- 
tions similar  to  the  preceding  in  relation  to  annular  wheels,  different 
cases  will  be  found  to  arise  under  varying  conditions,  the  treatment  of 
which  will  perhaps  be  best  developed  by  beginning  with  a  specific 
example. 


FIG.  135. 


THREE-LEAVED    PINION — INSIDE    GEAR. 


11? 


Lei  us  take  in  illustration  the  three-leaved  pinion,  Fig.   13G,  the 
conditions  being  as  follows  : 

Diam.  Inner  DCS.  Circle  =  f  Diam.  Pitch  Circle. 


Arc  of  approach.  45 c 


recess, 


75C 


Total  =  120°  =  Pitch. 


Eequired  to  find  the  least  annular  wheel  which  can  be  driven. 

Evidently  a  describing  circle  externally  tangent  to  both  pitch  cir- 
cles, as  in  Fig.  123, 
must  be  used  to  gener- 
ate the  faces  of  the  pin- 
ion and  the  flanks  for 
the  wheel;  and  its  least 
radius  may  be  found  as 
in  Fig.  133. 

Knowing,  then,  this 
radius  and  that  of  the 
pinion's  pitch  circle, 
for  the  reason  given  in 
(208)  the  radius  of  the 
larger  pitch  circle  must 
be  greater  than  their 
sum.  Also,  it  must  be 
such  that  the  pitch 
which  is  given  shall  be 
an  aliquot  part  of  the  circumference. 

Taking,  for  convenience,  5  as  the  radius  of  the  inner  describing 
circle,  then  A  C  —  8,  whence  by  computation  we  find  the  least  radius 
of  the  exterior  describing  circle  to  be  24.93. 

AD,  then,  must  be  greater  than  8  +  24.93  =  32.93  ;  let  it  equal  33 
for  instance. 

Now  the  numbers  of  the  teeth  are  proportional  to  the  radii  of  the 
pitch  circles  ;  and  letting 


FIG.  136. 


V  =  No.  teeth  of  Annular  Wheel, 
3  :  8  ::  N  :  33, 


we  must  have 

which  gives 

N  =  12|  • 

But  since  JV^must  be  an  integer,  the  least  number  that  can  be  used 
is  13  ;  and  the  corresponding  radius  of  the  pitch  circle  is  34f. 


118  INFERIOR  LIMIT  IK   INSIDE   GEAR. 

222.  With  this  radius  accordingly,  the  wheel  shown  in  Fig.  136  ia 
constructed  ;  the  radius  of  the  outer  describing  circle  is  taken  as  25, 
so  that  the  teeth  of  the  pinion  are  not  exactly  pointed,  though  their 
breadth  at  the  top  is  so  small  as  to  be  practically  inappreciable. 

And  a  glance  at  the  figure  is  sufficient  to  show  that  no  matter  how 
many  more  teeth  are  given  to  the  wheel,  the  pinion  will  work  with  it 
equally  well.  If  the  outer  pitch  circle  be  increased,  the  chord  of  the 
arc  AE  will  also  increase,  and  the  flank  PE  lying  always  outside  of 
the  face  PB,  will  continue  to  be,  as  now,  a  practicable  curve ;  which 
is  fcelf-evidently  true  in  regard  to  the  face  OF  oi  the  wheel-tooth. 

This  pinion,  it  will  be  observed,  is  precisely  the  same  as  the  one 
shown  in  Fig.  132  ;  and  is,  therefore,  capable  of  working  with  a 
rack,  or  with  a  wheel  -of  any  number  of  teeth  whatever,  between  the 
limits  of 

13  and  oo  in  Inside  Gear, 
oo  and  15  in  Outside  Gear. 

223.  From  this  example,  then,  we  have  thus  far  ascertained  that 
with  a  given  pinion,  under  certain  conditions  at  least,  there  is  an 
inferior  limit,  below  which  the  number  of  teeth  in  the  annular  wheel 
cannot  be  reduced,  but  no  superior  limit  beyond  which  the  number 
may  not  be  increased. 

To  these  results  we  call  attention,  the  more  particularly  because  so 
high  an  authority  as  Professor  Willis  makes,  without  further  com- 
ment, this  sweeping  statement,  viz.  : 

"  The  case  of  annular  wheels  differs  from  that  of  spur-wheels  in 
this  respect,  that,  with  a  given  pinion  a  small-numbered  wheel  works 
with  a  greater  angle  of  action  than  a  large-numbered  one,  and  there- 
fore we  have  to  assign  the  greatest  number  that  will  work  with  each 
given  pinion."  It  is  true  that  he  tacitly  recognizes  the  absence  under 
some  conditions  of  a  limit  in  that  direction,  by  assigning  "any  num- 
ber" as  the  greatest,  in  his  tables  of  limiting  cases;  but  neither  in 
this  fact,  nor  in  the  remarks  just  quoted,  is  there  any  recognition  of 
an  inferior  limit  under  any  conditions.  It  is  still  more  singular  that 
he  gives  nowhere  any  explanation  of  the  circumstances  under  which 
there  is  a  maximum  number,  nor  yet  of  the  only  mode  in  which  the 
teeth  can  possibly  be  generated  when  there  is  one  ;  neither  does  any 
other  writer  that  we  know  of. 

224.  Let  us  now  suppose  that  a  pinion  of  three  leaves  is  to  drive, 
under  the  conditions  that  the  tooth  shall  be  equal  to  the  space,  and 
the  arc  of  recess  just  equal  to  the  pitch. 


SUPERIOR   LIMIT   IIT   INSIDE   GEAR. 


119 


Then  in  Fig.  137,  A  0  being  the  diameter  of  the  pitch  circle,  we 
shall  have  on  its  circumference  the  space  A L  =  the  tooth  LB  =  60°  ; 
and  if  the  tooth  be  pointed,  its  point  must  lie  upon  M C,  perpendicu- 
lar to  CD,  the  line  of  centres.  Evidently,  if  we  draw  AL  and  OB, 
these  lines  produced  will  intersect  in  P  upon  MC,  making  OP  —  OA, 
and  the  arc  AP9  upon  the 
circle  whose  centre  is  0  and 
radius  OA,  will  be  equal  to 
the  arc  ALB.  That  circle 
may  therefore  be  used  as  a 
describing  circle,  and  by  roll- 
ing upon  the  pinion's  pitch 
circle  it  will  generate  the 
cardioidai  faces  PB  and  PL 
for  its  tooth.  This  describ- 
ing circle  cuts  CD  in  N ; 
drawing  PN  and  LO,  they 
will  be  parallel  to  each  other 
and  perpendicular  to  AP. 
Erect  a  perpendicular  to  PL 
at  its  middle  point  S ;  this 
will  pass  through  B,  and  bi- 
sect ON  in  D,  whence  FIG.  137. 


DA  :  OA  ::  3  :  2,    .  •.  DA  :  CA  ::  3  :  1  ; 


and  taking  D  as  the  centre  of  the  outer  pitch  circle,  an  arc,  AE,  of  40° 
upon  its  circumference  will  be  equal  to  the  arc  AP  of  60°  and  the  arc 
AB  of  120°. 

Bisect  AE  in  0,  and  draw  ED,  GD  ;  then  the  angle  included  be- 
tween these  radii  is  bisected  by  SD,  since  SDA  =  30° ;  consequently, 
L  is  the  highest  point  on  the  face  LG  of  one  tooth  of  the  annular 
wheel,  just  as  P  is  in  the  face  PE  of  the  next  one. 

225.  From  this  example  we  draw  these  conclusions,  viz. : 

1.  When  the  conditions  are  such  that  the  highest  point  in  the  face 
of  the  pinion's  tooth,  when  quitting  contact,  lies  above  the  common 
tangent  YY,  the  describing  circle  must  be  an  intermediate  one,  as  in 
Fig.  125. 

2.  The  intermediate  describing  circle  is  largest  when  the  tooth  is 
pointed. 

3.  The  larger  that  describing  circle,  the  larger  may  be  the  annular 


120  LIMIT   AFFECTED   BY   OBLIQUITY. 

wheel,  for  -(205)  AD  may  be  equal  to  AO  +  AC,  but  cannot  be 
greater. 

Now  by  the  construction  above  explained  we  have  ascertained  the 
maximum  diameters  of  the  describing  circle  and  of  the  outer  pitch 
circle,  the  latter  being  three  times  that  of  the  inner ;  therefore  9  is 
the  greatest  number  of  teeth  that  can  be  given  to  the  annular  wheel. 
The  number  assigned  by  Prof.  Willis  under  the  same  conditions 
is  12. 

226.  It  will  be  perceived  that  when  an  intermediate  describing  cir- 
cle is  used,  the  obliquity  varies  inversely  as  its  diameter.     If,  then,  a 
maximum  value  of  the  obliquity  be  assigned,  this  will  determine  a 
minimum  diameter  of  the  describing  circle,  and  consequently  of  the 
outer  pitch  circle,  which  fixes  in  this  case  also  an  inferior  limit  to  the 
number  of  teeth  in  the  annular  wheel.     Thus,  in  Fig.  137  it  is  clear 
that  although  if  we  reduce  the  outer  pitch  circle  we  may  also  reduce 
the  describing  circle,  we  cannot  doit  without  increasing  the  obliquity, 
of  which  the  maximum  value  in  the  figure  is  30°.     Now,  if  it  be  stip- 
ulated that  this  value  shall  not  be  exceeded,  we  must  retain  the  pres- 
ent describing  circle,  but  we  may  still  diminish  the  outer  pitch  circle. 
If  this  be  done,  the  face  PE  will  become  shorter,  and  the  limit, 
obviously,  will  be  reached  when  AD  becomes  equal  to  AO  =  2  AC. 
The  hypocycloid  PE  will  then  have  degenerated  into  the  point  P,  to 
which  the  whole  action  will  be  confined  ;  all  the  points  of  BP,  which 
remains  unchanged,  coming  successively  into  coincidence  with  P. 
We  find,  then,  that  under  this  additional  restriction  the  least  number 
of  teeth  that  can  be  employed  is  6.     If,  however,  it  should  under  any 
circumstances  be  desirable  for  the  modification  of  motion  to  use  such 
a  combination,  regardless  of  excessive  obliquity,  it  is  proper  to  note 
that  by  making  the  describing  circle  smaller  the  number  may  be  re- 
duced to  4 ;  and  in  general,  a  pinion  of  any  given  number  of  teeth 
may  thus  be  made  to  work,  more  or  less  satisfactorily,  with  an  annu- 
lar wheel  having  one  more,  a  fact  of  considerable  importance  in  the 
construction  of  differential  trains  of  wheels. 

227.  It  is  also  to  be  observed  that  in  Fig.  137,  the  point  L  upon 
the  inner  pitch  circle  is  at  once  the  highest  point  of  the  face  GL  and 
the  lowest  one  of  the  face  PL  ;  whence,  as  will  readily  appear  if  we 
suppose  the  motion  to  be  reversed,  the  arc  of  approach  is  equal  to 
half  the  pitch. 

Again,  the  face  A  Q  will  work  with  a  flank  generated  by  the  circle 
whose  radius  is  AH  =  OC  —  AC.  And  the  point  Q  lies  upon  the 
circumference  of  this  circle  ;  for  it  lies  on  PA  produced,  and  AQ  — 
AL  =  AC  =  AH,  whence,  if  HQ  be  drawn,  the  triangles  ACL, 


GENERAL   SOLUTION. 


AHQ,  will  be  equilateral  and  equal.  But  ACQ  —  ACT  —  30°  ;  so 
that  the  angle  QCT,  while  turning  through  which  the  pinion's  face 
has  two  points  of  driving  contact,  is  also  equal  to  half  the  pitch. 

In  rolling  the  outer  pitch  circle  upon  the  inner,  the  point  L  traces 
an  epifcrochoid,  to  which  AL  is  normal  and  LO  is  tangent  at  that 
point  (these  lines  corresponding  to  AR  and  R Fin  Fig.  124).  The 
outline  of  the  clearing  space  in  the  pinion  must  be  such  that  L  can 
move  freely  within  it,  but  may  also  be  tangent  to  LO  at  L.  But  the 
face  PL  is  tangent  at  the  same  point  to  the  radius  LC r;  consequently 
the  tooth  must  be  formed  with  a  positive,  although  an  obtuse,  inter- 
section at  that  point.  Still,  were  such  a  wheel  and  pinion  made  of 
the  proper  material  and  finish,  as  for  instance  of  hardened  steel  finely 
polished,  the  combination  might  be  used  in  light  mechanism. 

228.  The  above  is  a  special  case,  in  which  the  argument  is  based 
upon  peculiarities  of  the  assigned  conditions  ;  and  was  selected  because 
it  seemed  most  simply  and  clearly  to  illustrate  the  principles  involved. 

In  general,  however,  a  direct  geometrical  solution  like  this  is  not 
possible,  and  the  required  results  are  reached,  as  in  outside  gearing, 
by  the  process  explained  in  connection  with  Fig.  133.  In  adapting 
it  to  the  case  of  inside  gearing,  the  diagram,  as  shown  in  Fig.  138,  is 
modified  only  in  this  respect,  that  OA, 
and  consequently  the  centre  E  of  the 
intermediate  describing  circle,  will  lie 
above  instead  of  below  the  tangent  line 
A  G.  In  this  case  it  is  necessary  to  pro- 
duce CO  and  AG  in  order  that  they 
may  intersect  in  H ;  which  being  done, 
we  have  before  us  all  the  data  for  trigo- 
nometrical verification. 

If  the  obliquity  be  assigned,  the  con- 
struction is  made  in  the  following  order : 
Draw  the  indefinite  line  AQ,  making 
with  AG  an  angle  equal  to  the  given 
obliquity ;  then  the  arc  GO  cuts  AQ  in 
0,  the  highest  point  of  the  tooth.  Then 
as  before  bisect  OA  by  a  perpendicular 
cutting  AD  in  E ';  this  determines  AE,  FlG-  138- 

the  radius  of  the  least  intermediate  describing  circle. 

229.  The  conditions  might  be  such  that  the  point  0  should  fall 
exactly  upon  the  common  tangent.     In  this  event  OA,  equal  to  OF, 
is  itself  the  arc  of  the  describing  circle,  whose  radius  AE,  whether 
we  choose  to  regard  it  as  a  maximum  or  a  minimum,  is  infinite.    The 


122 


TWO-LEAVED   INTERNAL   PINION. 


pinion,  then,  can  just  work  with  a  rack,  but  if  the  tooth  be  pointed 
it  can  drive  nothing  less  ;  and  the  face  will  be  an  involute  of  the 
pitch  circle,  precisely  as  in  Fig.  121.  Since,  however,  the  tooth  need 
not  be  pointed,  the  pinion,  with  the  same  pitch  and  arc  of  recess,  can 
be  made  to  drive  any  annular  wheel  which  has  at  least  one  more  tooth 
than  itself,  by  the  use  of  an  intermediate  describing  circle  (see  226). 
230.  Two-leaved  Pinions  in  Inside  Gear. — An  annular  wheel  can  be 
driven  by  a  pinion  of  only  two  leaves,  the  teeth  working  in  the  same 
plane,  as  shown  in  Fig.  139.  Let  AC,  AO,  AD,  be  to  each  other  in 
the  proportion  of  1,  2,  3  ;  then  the  intermediate  describing  circle  is 
of  maximum  diameter.  Let  the  arc  of  recess,  AB,  =  135°,  then  the 
angle  AOB  =  67J°,  and  prolonging  OB  to  P,  the  arcs  AB,  AP,  will 


FIG.  139. 

be  equal ;  also  P  will  be  the  highest  point  of  the  cardioidal  face  PB, 
and  of  the  hypocycloidal  face  PE.  Draw  AP,  cutting  the  inner 
pitch  circle  in  R ;  then  RO  and  PU  will  be  perpendicular  to  A  P. 
Draw  D8  also  perpendicular  to  AP  ;  then,  since  A  U  is  bisected  at 
0,  and  OUatD,  PR  is  also  bisected  at  S. 

Therefore,  DP  =  DR,  that  is  to  say,  the  path  of  P  will  cut  the 
inner  pitch  circle  in  R.  If  then  we  suppose  the  motion  to  be  reversed, 
the  root  L  of  the  pinion's  face  LM  must  have  met  the  point  N  of  the 
wheel's  face  NG,  at  R.  In  other  words,  the  arc  of  approach  is  equal  to 
AP,  which  again  is  equal  to  RB,  because  R  O  bisects  the  angle  A  OP. 

231.  The  total  arc  of  action,  then,  is  ample,  that  of  ap- 
proach being  f,  and  that  of  recess  f ,  of  the  pitch  ;  and  the 
maximum  obliquity  being  37J°,  the  combination  might  be  used 


TWO-LEAVED    IKTEEKAL   PI^TIOK. 


123 


in  light  mechanism.  But  attention  is  called  to  a  peculiarity 
of  the  action  which  would  require  a  modification  in  the  finish 
of  the  teeth  for  practical  purposes.  When  contact  is  just  end- 
ing at  P  between  the  fronts,  or  acting  faces,  of  one  pair  of  teeth,  the 
backs  of  the  next  pair  are  just  coming  into  contact  at  /.  Now,  were 
the  teeth  of  the  pinion  merely  "topped  off'*'  in  the  usual  way  in  the 
lathe,  there  will  bo  danger  that  the  points  would  catch  upon  each 
other  near  the  point  I,  in  case  of  any  inaccuracy  of  workmanship  or 
wear  in  the  bearings.  This  risk  might  be  obviated  by  allowing  some 
backlash,  but  this  is  highly  objectionable  in  mechanism  of  the  only 
description  for  which,  if  for  any,  this  combination  is  suitable.  A 
much  better  expedient  in  such  cases  is  shown  in  the  figure,  the  top 
of  the  tooth  being  bounded  by  a  circular  arc  whose  centre  is  the  in- 
tersection of  the  normals  PA,  MO.  The  action  still  ends  at  P,  but 
the  other  tooth  of  the  pinion  will  have  been  guided  into  its  space  with- 
out risk  of  jamming.  And  it  may  be  added  that  a  similar  finish  of 
the  wheel's  tooth,  by  lengthening  it  a  little  and  rounding  off  the 
corner,  is  advisable  in  order  to  prevent  its  catching  upon  the  root  of 
the  pinion's  tooth,  since,  as  explained  in  (227),  there  will  be  a  blunt 
angle  at  L. 

The  tooth  of  the  pinion  here  shown  has  a  considerable  breadth, 
PM,  at  the  top,  and  it  will  be  seen  that  the  action  might  have  been 
extended  ;  also,  that  a  larger  pitch  circle  might  have  been  used  for 
the  wheel ;  but  as  will  sub- 
sequently appear,  the  max- 
imum number  of  teeth 
which  can  be  used  with  this 
pinion  is  seven. 

232.  A  pinion  with  two 
leaves  in  different  planes  may 
also  be  made  to  drive  an  an- 
nular wheel,  as  shown  in  Fig. 
140.  The  velocity  ratio  be- 
ing as  four  to  one,  let  A  C 
—  1,  AD  —  4  ;  then  the 
limiting  value  of  AH  =  3,  and  the  pinion  is  constructed  exactly  as 
in  Fig.  135,  being  in  fact  identical  with  the  one  there  shown,  which 
is,  therefore,  capable  of  driving  wheels  of  any  number  of  teeth  be- 
tween the  limits  of 


FIG.  140. 


6  and  oo  in  Outside  Gear. 
4  and  oo  in  Inside  Gear. 


124: 


ANGULAR   DRIVER   GIVEX. 


The  tooth  of  the  wheel  should  be  finished  as  shown  in  clotted  line  at 
EW,  by  which  the  injurious  action  of  a  sharp  corner  maybe  avoided  ; 
and  the  movement  is  very  smooth  and  noiseless,  while  the  amount  of 
sliding  is  much  less  than  in  outside  gear. 

233.  In  the  cases  thus  far  considered  the  pinion  has  been  the  driver  ; 
let  it  now  be  required  to  determine  the  least  number  of  teeth  that  can 
be  given  to  a  pinion  which  is  to  be  driven  by  an  annular  wheel  whose 
diameter,  pitch  and  arc  of  recess  are  assigned.  In  Fig.  141  let  D  be 
the  centre  of  this  wheel,  ^jPthe  arc  of  recess,  FL  the  half  thickness 
of  the  tooth,  whose  face,  evidently,  must  be  generated  by  an  interior 
describing  circle.  On  the  tangent  at  A,  set  off  A  G  =  arc  AF,  also 


FIG.  142. 

AM  —  \  A  G  ;  with  centre  M  and  radius  M G  describe  an  arc  cut- 
ting DL  in  0  ;  draw  OA,  and  bisect  it  by  a  perpendicular  cutting  AD 
in  E.  Then  E  is  the  centre  and  EA  the  radius  of  the  least  interior 
describing  circle,  and  if  the  pinion's  flanks  are  to  be  radial  the  mini- 
mum radius  of  its  pitch  circle  will  be  AC  —  2  AE\  but  if  we  adopt 
the  limit  mentioned  in  (210)  it  will  be  A  C  =.  -f  AE.  Since, 
however,  AE  cannot  exceed  CD,  it  follows  that  if  its  value  as  above 
determined  prove  to  be  greater  than  1  AD,  in  the  one  case,  or  T5?  AD 
in  the  other,  the  assigned  conditions  cannot  be  satisfied. 

If  a  maximum  obliquity  be  assigned,  the  teeth  may  or  may  not  be 
pointed,  and  the  diagram  is  constructed  by  first  drawing  AQ.  making 
with  A  G  an  angle  equal  to  the  given  obliquity,  and  then  describing 


FOLLOWER   GIVEN — OUTSIDE   GEAR. 


125 


the  arc  about  M,  to  cut  A  Q  in  0,  which  will  be  the  highest  point  of 
the  wheel's  tooth -face,  whether  the  tooth  itself  be  blunted  or  not. 

234.  Low- numbered  Pinions.  Follower  Given. — A  new  phase  of  the 
question  presents  itself  when  the  assigned  conditions  relate  to  the 
follower,  and  the  least  number  of  teeth  for  the  driver  is  to  be  de- 
termined. This  requires  a  different  mode  of  operation,  which,  as 
before,  will  be  best  explained  by  first  considering  a  case  in  outside 
gear. 

In  Fig.  142,  let  C  be  the  centre  of  the  follower;  then  A K,  the 
assigned  arc  of  recess,  is  equal  to  the  arc  A  0  of  a  describing  circle 
whose  radius  AE  is  known,  and  0  will  be  the  highest  point  of  the 


UNIVERSITY 


FIG.  143.  FIG.  144. 

driver's  tooth.  Let  D  be  the  centre  of  the  driver  ;  then  AD  will  be 
a  minimum  when  the  tooth  is  pointed  as  in  the  figure,  in  which  case 
OD  bisects  and  is  perpendicular  to  the  chord  FG  subtending  the  tooth. 
Draw  a  parallel  to  FG  through  A,  cutting  the  driver's  pitch  circle  in 
P\  then  the  chord  A P  is  also  bisected  by  OD,  and  AO,  PO,  are 
equal.  We  have  also  AF  =  PG,  and 


AP  =  AF  +  PG  -  FG,  = 


-  FG, 


AP  =  2  (arc  of  recess)  —  (thickness  of  tooth). 
Hence  the  construction,  Fig.  143,  in  which  A  0  is,  as  before,  the 


as- 


126  FOLLOWER   GIVEN  —  INSIDE   GEAR. 

signed  arc  of  recess  laid  off  on  the  describing  circle  of  the  follower's 
flank.     Draw  a  tangent  to  this  arc  at  A,  on  which  set  off 

AO  =  2  (arc  A  0)  —  (thickness  of  tooth), 
also 

AM= 


About  centre  M  with  radius  M  G  describe  the  indefinite  arc  GK  ; 
about  centre  0  with  radius  OA  describe  another  arc  cutting  GK  in 
P.  Draw  AP,  and  bisect  it  by  a  perpendicular  which  will  pass 
through  0  and  cut  CA  produced  in  /),  then  AD  will  be  the  minimum 
radius  of  the  driver. 

235.  Should  the  point  P  fall  upon  the  tangent  line,  the  assigned 
conditions  can  just  be  satisfied  by  a  rack  with  pointed  teeth,  but  by 
nothing  less  in  outside  gear. 

It  may,  however,  fall  upon  the  same  side  of  the  tangent  with  the 
centres  (7  and  E,  as  in  Fig.  144.  In  that  case  the  driver  will  be  an- 
nular, and  its  radius  AD  will  be  a  maximum  when  the  tooth  is 
pointed.  But  A  E  cannot  in  any  event  exceed  CD  ;  consequently,  if 
AD  prove  to  be  less  than  *-£-  AC  (or  f  AC  if  the  follower  have  radial 
flanks)  the  assigned  conditions  cannot  be  satisfied. 

If  it  prove  to  be  greater,  the  maximum  value  may  be  used  or  not, 
at  pleasure,  for  the  tooth  need  not  be  pointed,  and  by  topping  it  off, 
as  shown  in  Fig.  145,  the  radius  of  the  outer 
pitch  circle  may  be  reduced,  but  not  below  the 
limit  just  named. 

It  appears,  then,  that  with  a  given  pinion  there 
may  or  may  not  bo  a  superior  limit  to  the  num- 
ber of  teeth  for  the  annular  driver,  but  in  either 
case  there  is  an  inferior  one  if  there  be  any  re- 
ceding action  ;  the  wheel  cannot  have  less  than 
one  and  a  half  times  as  many  as  the  pinion,  when 
the  latter  has  radial  flanks. 

236.  One  case  remains  to  be  considered,  viz.,  when  the  given  fol- 
lower is  annular  ;  in  regard  to  which  it  has  already  been  pointed  out 
(226)  that  in  general  the  wheel  may  be  driven  by  a  pinion  having  one 
tooth  less  than  itself,  which  may  be  called  a  natural  maximum. 

By  reference  to  Fig.  129,  it  will  be  seen  that  the  larger  the  pinion 
the  broader  will  the  teeth  be  at  the  top,  and  that  its  diameter  will  be 
a  minimum  when  its  teeth  are  pointed.  Now  in  all  the  previous  cases 
the  assigned  conditions  have  enabled  us  to  fix  the  position  of  the 
highest  point  of  the  tooth  at  the  instant  of  quitting  contact.  This, 


ANNULAR   FOLLOWER   GIVEN.  127 

/ 

however,  is  not  so  in  the  present  case,  and  it  is  therefore  impossible 
to  determine  that  minimum  by  direct  means. 

But  it  may  be  found  indirectly  ;  for  if  we  ascertain  the  limiting 
numbers  of  teeth  for  the  annular  wheels  driven  by  various  given  pin- 
ions, we  shall  know  from  mere  inspection  of  these  results,  the  least 
number  for  the  pinion  which  can  be  used  to  drive,  under  like  condi- 
tions, in  inside  gear,  a  wheel  of  any  assigned  number  of  teeth. 


CHAPTER  VIII. 


SPUB   GEAKING,    CONTINUED — EPICYCLOIDAL   SYSTEM. 


Limiting  Numbers  of  Teeth  for  Various  Arcs  of  Action.  Details  of  Trigonomet- 
rical Process  of  Determination.  The  Nomodont,  or  Curve  of  Limiting 
Values. 

Computation  of  Tables. 

237.  It  can  always  be  determined  by  construction,  whether  a  proposed 
pair  of  wheels  will  work  under  given  conditions  as  to  pitch,  arc  of 
action,  etc.    But  in  order  to  avoid  wasting  time  by  attempting  impos- 
sible cases,  it  is  well  that  the  limiting  numbers  of  teeth,  within  a 
reasonable  range  of  varying  conditions,  should  be    ascertained  and 
tabulated  for  reference. 

Prof*  Willis,  in  his  "Principles  of  Mechanism,"*  gives  a  diagram 
and  deduces  from  it  an  expression,  confessedly  "so  involved  as  to 
make  the  direct  solution  of  the  equation  impossible,  although  approx- 
imations may  be  obtained."  He  adds  :  "  However,  on  account  of  the 
practical  importance  of  the  question,  I  have  arranged  in  the  following 
Tables  the  exact  required  results,  which  I  derived  organically  from 
the  diagram  by  constructing  it  on  a  large  scale  with  movable  rulers." 

This  rather  obscure  expression  evidently  indicates  the  use  of  an  ad- 
justable mechanical  device  of  some  kind  ;  but  since  this,  whatever  it 
was,  must  have  been  capable  of  being  drawn  in  any  phase  of  its  move- 
ments, the  inference  is  a  fair  one  that  his  process  was  equivalent  to 
that  of  deducing  results  graphically  by  trial  and  error. 

238.  Prof.  Willis  may  perhaps  have  overestimated  the  practical  im- 
portance of  the  question,  since  limiting  cases  are  in  general  to  be 
avoided  when  possible.      Still,  the   discussion  has  developed  some 
hitherto   overlooked  points  of  considerable   abstract   interest;   and 
again,  the  practical  value,  whatever  it  may  be,  of  such  tables,  depends 

*2d.  Ed.  1870,  pp.  106,107. 


COMPUTATION   OF  TABLES.  129 

entirely  upon  their  correctness.  And  by  the  application  of  Prof. 
Rankine's  singularly  elegant  processes  relating  to  circular  ares,  as  ex- 
plained in  the  preceding  paragraphs,  we  have  been  enabled  to  deter- 
mine these  limiting  numbers  accurately,  we  believe  for  the  first  time. 
For  the  numbers  in  the  tables  above  mentioned  are  those  of  teeth  in 
complete  circular  wheels,  capable  of  transmitting -rotation  continuously 
in  the  same  direction  ;  and  though  the  processes  of  Prof.  Rankine  are 
only  approximations,  they  are  such  close  ones  as  to  preclude  the  pos- 
sibility of  any  error  in  the  computations  arising  from  this  fact,  of 
such  magnitude  as  to  affect  the  integers  in  the  final  results. 

Prof.  Willis  calls  particular  attention  to  the  statement  that  his 
tables  are  "geometrically  exact."  *  It  has  already  been  demonstrated 
(224,  225)  that  in  at  least  one  instance  this  is  very  far  from  being  the 
case.  And  the  results  obtained  by  the  above  methods  differ  from  his 
in  so  many  other  instances,  and  so  widely,  that  in  view  of  his  high 
standing  as  an  authority,  we  feel  called  upon,  before  presenting  our 
own  Tables  of  Limiting  Numbers,  to  illustrate  in  detail  the  manner 
in  which  the  values  there  assigned  were  computed. 

239.  As  the  first  example  we  select  the  following  case. 

p .          j  Driving  Pinion  of  5  leaves,  Radial  Flanks. 
''  \  Arc  of  Recess  =  Pitch.     Tooth  =  Space. 

The  numbers  of  teeth  being  directly  proportional  to  the  radii,  let 
rad.  pinion  =  5  ;  constructing  the  diagram,  we  find  the  point  0  to 
lie  in  relation  to  AG  as  in  Fig.  138,  showing  that  the  pinion  cannot 
under  these  conditions  drive  a  wheel  in  outside  gear,  and  in  that  figure 
we  have 

ACF  =  72°,  arc  AF  =  }  x  5  x  2  x  3.1416  =  6.2832 
ACH=  54°,  AG  =  arc  AF  =  6.2832 

AHC=  36°,  AM  =   i  AG  =  1.5708 

AC  =  5,  OM  =   |  AG  =  4.7124 


Rad.  10. 

tan  ACff=  54°  10.138739 

A  C  =  5.  0.698970 

All  =  6.8819  0.837709 
AM—  1.5708 
HM  =  5.3111 

*  Principles  of  Mechanism,  p.  110. 


130  COMPUTATION  OF  TABLES. 

(OM  =  4.7124  ar.  com.     9.326758 

:  HM=  5.3111  0.725183 

::  sin  OHM  =    36°  9.769219 

:  sin  HOM  =  138°  30'  44"  9.821160 
OMA  =  174°  30'  44" 

180° 

0AM  +  AOM  =      5°  29'  16";  -f-    2    =  2°  44'  38" 


OM  +  AM  =  6.2832  ar.  com.     9.201819 

Triangle  j     :  QM  -  AM  =  3.1416  0.497151 

::  tan.  £  (A  +  0)  =  2°  44'  38"  8.680580 

:  tan  $  (A  —  0)  =  1°  22'  22"  8.379550 

0AM          =  ¥~1'       =  Max.  Obliq.  =  AEN. 

(       siuOAM=      4°     7'  ar.com.     1.143951 

Triangle    I     :  sin  QMA  =  174°  30'  44"  8.980609 

OM=  4.7124  0.673242 

OA  0.797802 

2,  0.301030 

=  \  OA  0.496772 


sin  AEN '=  4°  7'  ar.  com.       1.143951 

Triangle    I     :  AN  0.496772 

AEN.     I    ::  Rad.  10. 

:  AE  —  43.7343  =  Rad.  Interm.  Des.  Circle.  1.640723 

AC  =     5.  (85  =  Max.  No.  given  by  Prof.  Willis.) 

48.7343  .-.  48-  Maximum  No.  Teeth. 

240.   We  will  next  take  a  case  in  outside  gear,  thus  : 


Given    -S  Follower  of  10  Teeth.     Eadial  Flanks. 

'  1  Arc  of  Recess  =  Pitch.     Tooth  =  Space. 

Then  in  Fig.  143  we  shall  have 

AG  =  |  x  10  x  TV  x  2  x  3.1416  =  9.4248. 

180C 

0AM  =  36( 

AOM  +  AMO  =       Iff 
i  (M  +  0)  =  72< 


AM  =  ±AG  =  2.3562 


PM  =  IAG  =  7.0686 
AC  =  10  =  Rad.  Follower. 


ACM  =   OAM  =  3Q°=  Max.  Obliquity. 


COMPUTATION   OF  TABLES. 


131 


Triangle 
CAM. 


Triangle 
0AM. 


Bad. 

sin  0AM  = 

36° 

AC  =  10 

OA  =  OP 

=  5.8778 

AM 

=  2.3562 

OA  +  AM         =  8.2340 
:    OA  -  AM        =  3.5216 

::  tan  J  (M  +  0)  =  72° 

:  tan  J-  (M  -  0)  =  52°  46'  32" 
AOM  =  19°  13' 28" 
AOM=  9°  36' 44" 


10. 

9.769219 
JL 

0.769219 


ar.  com.      9.084389 

0.546740 

10.488224 

10.119353 


Triangle 
0AM. 


sin  AOM  =  19°  13'  28" 
sin  OA  M  =  36° 
AM  =  2.3562 
OM  =  4.2061 
OP  =  5.8778 
PM^  7.0686 

2)  17.1525  =  OM  +  OP  +  PJf. 
8.5763  =  8. 
1.5077  =  8  -  PM. 


ar.  com.     0.482449 


Triangle 
POM. 


cos 

,  /fiad2  .  S 

.   (S  —  PM} 

~Y     OP 

x   0Jf 

'Bad. 

9 

8 

=  8.5763 

s  — 

PJf  =  1.5077 

OP 

=  5.8778 

0.769219) 

OM 

=  4.2061 

0.  623880  f 

.  cos  -J 

P03f  =     43°  40'  51" 

J 

AOM—       9°  36'  44" 

AON=  53°  17'  35" 


OAD  =  90°  +  0AM  =   126 


9.769219 
0.372212 

0.623880 


20. 

0.933300 

0.178315 
•  21.111615 

1.393099 
2)  19.718516 

9. 859258 


ADN  =  180°  -  (179°  17'  35")  =  0°  42'  25" 

f       sin  ADN=    0°  42'  25"  ar.  com.     1.908777 

Triangle  j     :  sin  AON  =  53°  17'  35"  9.904014 

ADN.          ::OA=      5.8778  0.769219 

[    :  AD=  381.9531  2.582010 

.  •.    382  =  Minimum  No.  for  Driver. 


132 


COMPUTATION   OF   TABLES, 


241.  Prof.  Willis  correctly  observes  that  when  the  action  begins  at 
the  line  of  centres,  no  pinion  of  less  than  ten  leaves  can  be  driven,  but 
in  reference  to  the  above  case  he  says  (p.  210)  "nothing  less  than  a 
rack  can  drive  a  pinion  of  ten  ; "  whereas  it  appears  from  these  figures 
that  it  can  be  driven  by  any  wheel  of  382  teeth  or  more.  In  corrob- 
oration  of  which  we  will  now  assume  a  driver  of  382  teeth,  the  arc  of 
recess  being  equal  to  the  pitch,  and  the  tooth  to  the  space,  as  before. 

We  shall  then  have  in  Fig.  133, 


ACF  =      0°  56' 32.67" 
ACH  =      0°  42'  24.5" 
ARC  =    89°  17'  35.5" 
OHM  =     90°  42'  24.5" 
AC=  382. 


arc  AF  =  ^  x  382  x  2  x  3.1416  =  6.2832 
AG  =  arcAF=  6.2832 
AM  =    J   AG  =  1.5708 
OM  =   f    AG  =  4.7124 


Rad. 

Triangle    I     :   t&n  ACff  =  0°  42'  24.5' 
ACH.      I     ::  AC  =  382 

:  AH  =  4.7123 
AM  =  1.5708 
HM=  3.1415 


10. 

8.091169 
2.582063 
0.673232 


Triangle 
OHM. 

I 


OM-  4.7124 
HM=  3.1415 
sin  OHM  r-.    90°  42'  24.5" 
smHOM  =    41°  48'  17.5" 
OMA  =  132°  30'  42" 
180° 


ar.  com.  9.326758 
0.497137 
9.999967 
9.823862 


0AM  +  AOM  =    47°  29'  18"  ;  +  2  =  23°  44'  39" 


OM  +  AM  =  6.2832 
Triangle    I     :    QM  -  AM  =  3.1416 
OMA.     1     ::  tan  $  (A  +  0)  =  23°  44'  39" 
:  tan  |  ( A  -  0}  =  12°  24'    5.8 
0AM  = 


ar.  com.  9.201819 
0.497151 
9.643243 


9.342213 


36°    8'  45"  =  Max.Obliq.  =  AEN. 


sin  0AM  =    36°  8'  45" 
Triangle    I     :  sin  OMA  —  132°  30'  42 
OMA.     I     ''-  OM  =4.7124 
:   OA 
2 
AN  =      OA 


ar.  com. 


0.229236 
9.867596 
0.673242 
0.770074 
0.301030 
0.469044 


COMPUTATION   OF  TABLES.  133 

sin  AEN  =  36°  8'  45"                     ar.  com.     0.229236 

Triangle    I     :   AN  0.469044 

AEN.     1     "Rad.  10. 

:    AE  =  4.9921  —  Had.  Ext.  Des.  Circle.  0.698280 

2  (Flanks  to  be  Radial). 
9.9842  . '.  10  =  Minimum  JSTo.  for  Follower. 


242.  In  the  case  of  a  driving  pinion  of  6  leaves,  under  the  same 
conditions,  we  find,  by  a  similar  computation, 


AE  =    71.7083  =  Kad.  Exterior  Des.  Circle. 
Add  AC  =      6.          —  Ead.  Driver, 

then  77.7083  =  Rad.  Annular  Wheel, 

also  2  AE  =  143.4166  =  Rad.  Follower  in  Outside  Gear. 


These  being  minimum  values,  the  next  higher  integers  must  be 
taken  as  the  limiting  numbers  of  teeth  for  complete  wheels.  Thus 
the  6-leaved  pinion  can  drive  an  annular  wheel  of  78  teeth,  or  an  ex- 
ternally toothed  one  of  144,  but  no  less  numbers  can  be  used  when  the 
describing  circle  thus  found  is  employed. 

Prof.  Willis  assigns  176  instead  of  144,  and,  as  previously  stated,  he 
gives  a  rack,  instead  of  382  teeth,  as  the  least  that  can  drive  a  pinion 
of  10.  In  the  table  for  outside  gear  we  have  merely  marked  with  an 
asterisk  those  of  our  own  values  which  differ  from  his,  because  the 
above-mentioned  discrepancies  are  the  most  serious  ones.  But  his 
method,  whatever  it  was,  appears  to  have  been  singularly  defective  in 
its  application  to  inside  gear  ;  he  not  only  ignores  entirely  the  mini- 
mum values,  but  assigns  maximum  values  which  differ  so  often  and 
so  widely  from  those  computed  as  above,  that  we  have  deemed  it 
proper  to  present  his  tables  for  annular  wheels  side  by  side  with  our 
own,  which  we  think  it  safe  to  assert  are  numerically  exact.  Had  the 
object  been  to  determine  the  limiting  radii  with  the  utmost  precision, 
a  correction  for  the  error  in  Prof.  Rankine's  approximations  would  have 
been  necessary  in  every  instance.  This,  however,  we  have  applied 
only  when  the  arc  to  be  dealt  with  was  so  large  as  to  make  it  requisite, 
in  order  to  secure  in  the  computation  of  converse  cases,  as  in  (240) 
and  (241),  results  concordant  in  respect  to  the  number  of  teeth  for 
complete  wheels. 


134 


OUTSIDE  G  EASING. 

Limiting  Numbers  of  Teeth. 


BECESS  =  PITCH. 

RECESS  —  J  PITCH. 

RECESS  =  f  PITCH. 

D. 

F. 

D. 

F. 

D. 

F. 

5.58 

oo 

3.25 

00 

2.58 

00 

6 

144* 

4 

34* 

3 

37* 

7 

50* 

5 

19 

4 

15 

8 

34* 

6 

14 

5* 

11* 

9 

27 

7 

12 

6 

10 

10 

23 

8 

11* 

7 

9 

11 

21 

9* 

10 

8 

8 

12 

19 

11* 

9* 

11 

7 

13 

18 

16 

8 

21* 

6 

14 

17 

33* 

7 

oo 

5.17 

15 

16 

00 

6,35 

17 

15 

20 

14 

24 

13 

32* 

12 

OUTSIDE  GEARING. 

EPICYCLOIDAL  TEETH.     RADIAL  FLANKS. 

57* 

11 

Tooth  =  Space. 

382* 

10 

Minimum   Values. 

00 

9.83 

*  Numbers  differing  from  those  given  by  Prof.  Willis. 


INSIDE   EPICYCLOIDAL   GEARING. 

Limiting  Numbers  of  Teeth. 


135 


INSIDE   EPICYCLOIDAL   GEARING. 

FLANKS  OF  PINIONS   RADIAL.      TOOTH  =  SPACE. 

RECESS  =  PITCH. 

EECESS  =  J  PITCH. 

RECESS  =  |  PITCH. 

D. 

F. 

D. 

F. 

D. 

F. 

3 

9 

2 

7 

2 

11 

4 

16 

3 

41 

3 

CO 

5 

48 

4 

oo 

6 

00 

Maxim  um  Followers. 

F. 

D. 

F. 

D. 

F. 

D. 

7 

14 

5 

12 

4 

8 

8 

24 

6 

65 

5 

53 

9 

60 

7 

00 

6 

00 

10 

00 

Maximum  Drivers. 

D. 

F. 

D. 

F. 

D. 

F. 

6 

78 

4 

21 

3 

22 

7 

32 

5 

15 

4 

12 

8 

25 

6 

13 

5 

11 

9 

23 

7 

13 

6 

11 

10 

22 

8 

14 

7 

12 

11 

22 

9 

14 

8 

12 

12 

22 

10 

15 

9 

13 

13 

22 

11 

16 

10 

14 

14 

23 

12 

17 

11 

15 

15 
16 

23 
24 

Minimum  Followers. 

136 


INSIDE   EPICYCLOIDAL   GEARING. 

Limiting  Numbers.     (Prof.    Willis.) 


INSIDE   EPICYCLOIDAL   GEARING. 

FLANKS   OF  PINIONS   BADIAL.      TOOTH  =  SPACE. 

RECESS  =  PITCH. 

RECESS  =  J  PITCH. 

RECESS  =  f  PITCH. 

D. 

F. 

D. 

F. 

D. 

F. 

2 

5 

2 

10 

2 

14 

3 

12 

3 

77 

4 

Any  No. 

4 

26 

4 

Any  No. 

5 

7 

85 
Any  No. 

Maximum  Followers. 

F, 

D. 

F. 

D. 

F. 

D. 

7 

14 

4 

5 

4 

8 

8 

25 

5 

12 

5 

64 

9 

60 

6 

77 

10 

Rack. 

Maximum  Drivers. 

243.  The  manner  of  using  these  tables  is  best  illustrated  by  an  ex- 
ample or  two.  In  outside  gearing,  let  the  given  wheel  be  a  driver  of 
12  teeth,  the  arc  of  recess  to  equal  the  pitch.  In  this  division  are 
two  columns,  marked  D  and  F,  for  drivers  and  followers  respectively  : 
and  opposite  12  in  column  D,  we  find  in  column  F  the  number  19, 
which  is  the  least  that  can  be  driven.  If  the  same  number  be  assigned 
for  a  follower,  we  find  in  column  D,  opposite  12  in  column  F,  the 
least  number,  32,  that  will  drive. 

If  the  given  number  be  not  found  in  the  table,  the  number  for  the 
required  wheel  will  be  found  opposite  the  next  less  number.  For  in- 
stance, the  arc  of  recess  being  three-fourths  of  the  pitch,  let  the  given 
wheel  have  28  teeth.  By  reference  to  the  table,  it  is  seen  that  16 


EXPLANATION   OF   TABLES.  137 

(or  over)  will  drive  8,  but  33  is  the  least  that  will  drive  7  ;  hence  the 
given  wheel  will  drive  8,  but  no  less.  The  given  number  is  not  found 
in  column  F  either  ;  but  19  can  be  driven  by  5,  or  any  number  greater 
than  five,  while  a  pinion  of  4  can  drive  no  less  than  34  :  consequently 
5  is  the  least  that  can  be  usecj  to  drive  the  wheel  of  28.  Thus  this 
table  includes  all  possible  limiting  numbers  for  the  three  values  of  the 
arc  of  recess,  in  outside  gearing. 

In  the  first  division  of  the  table  for  inside  gearing,  marked  Maxi- 
mum Followers,  the  columns  F  contain  the  greatest  numbers  of  teeth 
for  the  annular  wheels  which  can  be  driven  by  the  corresponding 
numbers  in  the  columns  D  ;  for  instance,  when  recess  =  pitch,  a  pinion 
of  5  can  drive  a  wheel  of  48,  but  not  more.  Since  the  least  number 
that  can  be  driven  is  alwa"ys  one  more  than  that  on  the  pinion,  it  is 
not  given  in  the  table  ;  it  is  necessary  only  to  remark  that  the  lowest 
numbers  which  can  be  thus  used  are  3  for  the  pinion  and  4  for  the 
wheel,  as  a  two-leaved  pinion  will  not  drive  on  account  of  the  exces- 
sive obliquity  when  the  wheel  has  but  three  teeth.  In  all  the  com- 
binations in  this  division,  the  receding  action  can  be  secured  only  by 
the  use  of  an  intermediate  describing  circle. 

In  the  second  division,  the  columns  D  contain  the  greatest  numbers 
for  the  annular  wheels  which  can  drive  the  corresponding  numbers  in 
columns  F\  thus  when.recess  =  f  pitch,  a  pinion  of  6  can  be  driven 
by  a  wheel  of  65  teeth,  but  not  by  a  larger  one.  Since  the  flanks  of 
the  pinion  are  to  be  radial,  the  least  number  for  the  driver  (235)  will 
be  one  and  a  half  times  that  of  the  given  pinion,  whatever  the  arc  of 
recess,  and  therefore  is  not  set  clown  in  the  table. 

In  the  third  division,  the  columns  D  contain  the  numbers  for  given 
pinions  whose  teeth  are  pointed,  the  faces  being,  therefore,  generated 
in  each  case  by  the  least  possible  describing  circle.  The  columns  F 
contain  the  least  numbers  for  the  annular  wheels  which  can  be  driven 
by  these  pinions,  whose  faces  act  against  flanks  traced  by  the  same 
describing  circles,  which  are  exterior  ones  ;  and  when  these  least  num- 
bers are  used,  the  annular  wheels  can  have  no  faces.  There  being, 
then,  no  approaching  action,  it  would  at  first  sight  appear  that  the 
action  could  only  be  continuously  maintained  when  the  arc  of  recess 
is  equal  to  the  pitch.  Bat  the  assigned  amount  of  receding  action  in 
these  cases  is  secured  by  the  use  of  the  exterior  describing  circle  only : 
and  the  actual  amount  is  greater,  owing  to  the  peculiarity  of  the  ac- 
tion explained  in  (206),  there  being  in  fact  two  points  of  driving  con- 
tact up  to  the  limit  assigned  in  the  table,  so  that  the  rotation  will  be 
properly  maintained  in  every  case. 

244.  It  remains  now  to  illustrate  by  an  example  the  use  of  this 


138  EXPLANATION   OF  TABLES. 

table  in  finding,  as  mentioned  in  (236),  the  limiting  number  of  teeth 
for  a  pinion  which  is  to  drive  a  given  annular  follower. 

Suppose  that  the  recess  is  to  be  equal  to  the  pitch  :  then,  recollect- 
ing that,  as  just  mentioned,  a  pinion  of  3  can  drive  a  wheel  of  4,  we 
deduce  at  once,  in  regard  to  the  least  number,  the  following  by  in- 
spection of  the  tablo  : 

No.  ON  GIVEN  WHEEL.  LEAST  No.  TOP.  PINION.  DESCRIBING  CIRCLE. 

4  -    9  inclusive 3    j 

nils   "  ::::::::::  J  intermediate. 

49-77         "        6   J 

78-oo  6        Exterior. 

The  greatest  number  for  the  pinion,  if  an  intermediate  describing 
circle  be  used,  is  always  one  less  than  the  given  number  on  the  annu- 
lar wheel. 

If  it  bo  stipulated  that  the  assigned  arc  of  recess  shall  be  secured 
by  the  use  of  an  exterior  describing  circle,  the  limiting  numbers  may 
also  be  found  by  simple  inspection  if  within  the  range  covered  by  the 
table.  But  in  order  to  include  the  superior  limit  for  even  moderately 
large  wheels,  the  table  would  require  to  be  inconveniently  extended. 
For  instance,  let  the  given  number  be  42  ;  this  is  found  by  computa- 
tion to  be  the  least  that  can  be  driven  by  a  spur-wheel  of  36  teeth, 
which  is  therefore  the  largest  possible  driver.  A  pinion  of  6  can 
drive  no  less  number  than  78,  but  one  of  7  can  drive  any  number 
above  32,  and  is,  consequently,  the  least  which  can  be  used  under  the 
assigned  conditions. 

245.  If  an  interchangeable  set  of  spur  wheels  be  constructed  by  the 
use  of  a  constant  describing  circle,  the  limiting  numbers  for  the  an- 
nular wheels  which  will  gear  with  them  may  be  readily  found  without 
reference  to  a  table.  In  the  system  formerly  employed  by  Brown  & 
Sharpe,  of  Providence,  K.  I.,  the  diameter  of  the  describing  circle  was 
equal  to  the  radius  of  a  pinion  of  twelve  teeth.  If  then  the  teeth  of 
the  annular  wheels  are  to  have  both  faces  and  flanks,  the  distance  be- 
tween centres  of  the  interior  and  exterior  describing  circles  may  be 
represented  by  twelve,  since  the  radii  are  proportional  to  the  numbers 
of  teeth  ;  and  at  the  limit  this  distance  is  equal  to  that  between  the 
centres  of  the  pitch  circles.  In  these  circumstances,  therefore,  any 
spur  wheel  of  the  set  will  gear  with  an  annular  wheel  having  twelve 
more  teeth  than  itself,  but  no  less  than  that  will  answer.  Either  may 
be  used  as  driver  or  as  follower  indifferently,  but  the  relative  and 
the  actual  amounts  of  approaching  and  receding  action  will  depend 


THE   NOMODOtfT.  139 

not  only  upon  which  does  drive,  but  also  upon  the  actual  numbers  of 
teeth  used  in  any  given  case.  Since,  however,  in  the  use  of  this  sys- 
tem the  smallest  pinion  ever  used  was  one  of  twelve  teeth,  the  total 
angle  of  action  would  clearly  be  always  ample. 

By  cutting  down  the  pinion  to  the  pitch  circle  we  may  use  a  smaller 
wheel  (as  a  driver  only),  and  by  similarly  treating  the  wheel  we  may 
use  a  smaller  pinion  for  the  same  purpose  ;  the  least  number  for  the 
annular  wheel  being  found  in  either  case  by  adding  six,  instead  of 
twelve,  to  the  number  of  teeth  upon  the  given  internal  wheel. 

In  the  system  first  adopted  by  Pratt  &  Whitney,  of  Hartford,  Conn., 
and  subsequently  by  Brown  &  Sharpe,  the  diameter  of  the  describ- 
ing circle  is  equal  to  half  that  of  the  pinion  of  fifteen  teeth.  These 
limiting  numbers  are,  therefore,  found  by  adding  to  the  number  of 
teeth  on  the  given  spur-wheel,  fifteen  if  both  wheels  are  to  have  faces 
as  well  as  flanks,  and  eight,  if  either  is  to  be  cut  down  to  the  pitch 
line  ;  in  the  latter  case,  the  exact  number  to  be  added  is  seven  and  a 
half,  which  giving  a  fractional  result,  the  next  higher  integer  must  be 
taken. 

246.  The  Nomodont. — If  the  radius  of  the  given  wheel  be  gradually 
increased,  other  things  remaining  unchanged,  it  is  obvious  that  the 
radius  of  the  greatest  or  the  least  wheel,  as  the  case  may  be,  which  will 
work  with  it  either  as  a  driver  or  a  follower,  will  vary  according  to 
some  regular  law.     As  in  many  other  cases,  this  law  can  be  best  illus- 
trated graphically,  by  a  curve  whose  abscissas  are  proportional  to  the 
given,  and  the  ordinates  to  the  required,  radii  of  the  wheels  ;  to  which 
we  have  given  the  descriptive  name  of  the  Nomodont,  from  the  law 
expressed  by  it  in  relation  to  the  numbers  of  teeth.     These  limiting 
numbers,  as  given  in  the  tables,  are  integers,  as  they  must  be  if  com- 
plete wheels  are  to  be  used  ;  this  would  not  be  the  case  were  sectors 
only  to  be  employed,  as  they  often  are,   and  in  constructing  these 
curves  the  fractional  values  as  computed  have  been  given  to  the 
ordinates. 

It  is  hardly  necessary  to  point  out  that  the  regularity  of  the  curves 
affords  a  quite  rigid  test  of  the  correctness  of  the  results  which  they 
embody.  And  in  order  to  define  them  the  more  perfectly,  especially 
between  the  origin  and  the  vertices,  in  the  region  of  the  asymptotical 
ordinates,  a  great  number  of  intermediate  fractional  radii  were  as- 
sumed for  the  given  wheels,  the  conditions  for  all  the  curves  repre- 
sented being,  that  the  externally  toothed  wheels  shall  have  radial 
flanks,  that  the  arc  of  recess  shall  be  equal  to  the  pitch,  and  the  tooth 
equal  to  the  space. 

247,  In  Fig.  146,  the  abscissas  represent  the  radii  of  given  exter- 


140 


THE   KOMODOKT. 


nally  toothed  wheels  which  are  to  drive,  the  ordinates  of  the  curves 

A  and  B  are  proportional  to  the  corres- 
ponding minimum  radii  of  the  followers 
in  outside  and  inside  gear  respectively, 
determined  as  in  the  case  of  the  6-leaved 
pinion,  in  (243). 

This,  as  appears  from  the  table,  is  the 
least  complete  pinion  which  can  drive 
at  all  in  outside  gear ;  the  exact  limit- 
ing radius  of  one  which  can  just  drive  a 


THE  NOMODONT. 

Epicycloidal  System.    Recess  =  Pitch.  Tooth  =  Space. 
Scale  of  Ordinatcs  j  Scale  of  Abscissas. 


ia  20  28 

RADII  OF  GIVEN  DRIVERS.    FIG.  146. 


rack,  is  5. 58,  at  which  point  the  ordinate  is  therefore  infinite,  and  asymp- 
totic to  the  curve  A.  As  the  radius  of  the  given  driver  increases,  that 
of  the  follower  diminishes,  at  first  very  rapidly,  then  more  and  more 
slowly,  until  the  driver  becomes  a  rack,  when  the  value  of  the  ordi- 
nate is  9.83,  which  agrees  with  the  tabular  record  that  the  least  whole 
number  which  can  be  driven  is  10  ;  consequently  this  curve  has  also 
a  horizontal  asymptote  at  a  distance  of  9.83  above  the  axis  of  ab- 
scissas. 

Since  the  driven  rack  may  be  regarded  as  an  infinitely  large  wheel 
in  inside  gear,  as  well  as  in  outside,  the  ordinate  at  5.58  is  also  an 
asymptote  to  the  curves  B  and  C ;  the  ordinates  of  the  latter  being 
proportional  to  the  maximum  radii  of  annular  followers,  whose  action 
requires  the  use  of  an  intermediate  describing  circle,  determined  as  in 
(239). 

The  ordinates  of  B  also  diminish  rapidly  at  first,  but  reaching  a 
minimum  when  the  radius  of  the  driver  is  between  10  and  12,  subse- 
quently increase.  The  rate  of  increase,  however,  is  not  constant ;  the 
value  of  the  ordinate  is  (239)  AC  +  AE,  in  which,  while  AC  in- 
creases uniformly,  AE  diminishes  more  and  more  slowly,  reaching  the 
limit  of  4.92  when  AC  =  <=*=>.  If,  then,  any  two  ordinates  be  drawn, 
each  measuring,  by  the  scale  of  ordinates,  a  distance  equal  to  its  abscissa 


THE   NOMODCWT. 


141 


+  4.92,  the  right  line  passing  through  their  extremities  will  be  a 
second  asymptote  to  the  curve  B. 

Both  A  and  B  are,  it  will  be  observed,  remarkably  symmetrical 
curves,  being  in  fact  very  nearly  true  hyperbolas.  The  ordinates  of 
(7,  on  the  other  hand,  diminish  very  rapidly  as  the  radius  of  the  driver 
is  reduced,  their  values,  as  shown  in  (239)  and  (225),  being  48.73 
when  that  radius  is  5,  and  exactly  9  when  the  radius  is  3.  As  ap- 
pears from  the  table,  only  three  complete  pinions,  having  respectively 
3,  4,  and  5  teeth,  are  included  in  the  class  here  represented.  Prof. 
Willis  gives  a  pinion  of  two  leaves,  as  capable  of  driving  a  wheel  of 
five  teeth  ;  but  this  will  -be  found  wholly  impracticable  on  account  of 
the  excessive  obliquity. 

We  have,  therefore,  not  considered  it  worth  while  to  compute  an 
ordinate  for  a  less  driving  radius  than  2.5,  at  which  limit  the  value 
is  6.76  ;  but  the  curve  is  extended  in  a  dotted  line  to  the  zero  point, 
on  the  assumption  that  driver  and  follower  would  vanish  simul- 
taneously. 

248.  In  Fig.  147  the  abscissas  represent  the  radii  of  given  externally 

toothed  followers.  The  ordi- 
nates of  the  curve  D  being  pro- 
portional to  the  corresponding 
minimum  radii  of  drivers  in 
outside  gear,  it  is  at  once  appar- 
ent that  this  curve  will  have  a 
vertical  asymptote  at  9.83,  and 
a  horizontal  one  at  a  distance  of 
5.58  above  the  axis  of  abscissas. 


THE  NOMODONT. 
Epicycloidal  System.  Recess=Pitch.   Teeth=Space. 


O.S3  2O  28  33 

RADII  OP  GIVEN  FOLLOWERS.    FIG.  147. 

The  curves  A  and  D  are  very  similar,  and  will  become  in  fact 
identical  if  the  axes  be  transposed  and  the  same  scale  used  for  both 
ordinates  and  abscissas  ;  for  it  will  readily  be  seen,  by  comparing  Fig. 
133  with  Figs.  142  and  143,  that  assuming  the  processes  of  construc- 
tion and  computation  to  be  exact  instead  of  approximative,  the  radii 
in  those  converse  cases  must  under  similar  conditions  be  precisely  the 


142  THE   NOMODOHT. 

same.  As  here  shown,  however,  the  curve  D  has  been  constructed 
with  ordinates  computed  as  in  (240). 

When  the  given  radius  is  less  than  9.83,  the  driver  must  be  annu- 
lar, and  its  maximum  radius  forms  an  ordinate  to  the  curve  E,  which 
is  somewhat  similar  to  (7,  and  like  it  includes  the  radii  of  but  three 
complete  pinions ;  which  have  respectively  7,  8  and  9  leaves. 

The  maximum  radius  of  the  driver  for  a  pinion  of  six  leaves,  is  7.45, 
which  is  impracticable,  as  we  have  seen  that  it  must  be  at  least  one 
and  a  half  times  as  great  as  that  of  the  follower.  Therefore  the  com- 
putations were  carried  no  further,  although  this  curve,  like  C,  and 
upon  the  same  assumption,  is  continued  in  a  dotted  line  to  the  zero 
point. 


UNIVERSITY 


CHAPTER  IX. 


SPUE   GEARING,  CONTINUED — INVOLUTE   TEETH. 


\r 


Involute  Generated  by  Rolling  of  Right  Line  on  Base  Circles.  Peculiar  Proper- 
ties. Original  Pitch  Circle.  Rack  and  Pinion.  Annular  Wheels.  Low- 
numbered  Pinions.  Involute  Tooth  with  Epicycloidal  Extension.  Limiting 
Numbers  for  Given  Arcs  of  Recess.  Comparison  of  the  Involute  and  Epicy- 
cloidal Systems.  Involute  Generated  by  Rolling  of  Logarithmic  Spiral  on 
Pitch  Circles. 

Involute  Teeth. 

249.  Next  to  the  epicycloid,  the  curve  most  extensively  used  as  the 
tooth-outline  in  spur  gearing,  is  the  involute  of  the  circle. 

It  will  subsequently  be  shown 
that  this  curve  can  be  generated 
in  a  manner  which  conforms  to 
the  general  law  enunciated  in 
(181) ;  but  its  fitness  for  this 
purpose  is  proved  much  more 
clearly  and  simply  by  deriving  it 
in  another  way. 

In  Fig.  148,  let  LM,  RS,  be 
portions  of  two  pitch  circles  in 
contact  at  A  ;  C  and  D  their  cen- 
tres, and  TT  their  common  tan- 
gent. Draw  NN  inclined  to  TT, 
and  upon  it  let  fall  the  perpen- 
diculars CE,  DF  \  with  which,  as 
radii,  describe  the  circles  EHY, 
FGQ.  Suppose  these  circles  to 
be  disks  upon  which  is  wound  an 
inextensible  thread  EF,  carrying 
a  marking  point  at  F.  Now  let  FlG-  148- 

the  upper  disk  turn  as  shown  by  the  arrow  :  it  will  wind  the  thread 
upon  itself,  unwinding  it  from  the  lower  one,  which  will  thus  be 


144  FIRST   GENERATION   OF  INVOLUTE. 

turned  in  the  opposite  direction,  but  with  the  same  perimetral  veloc- 
ity. Let  the  arcs  EH,  FG,  be  equal  to  each  other  and  to  the  right 
line  EF\  then  while  the  pencil  is  drawn  from  Fto  E,it  will  trace 
upon  the  plane  of  the  upper  disk  the  curve  KE,  and  upon  the  plane 
of  the  lower  one  the  curve  QE. 

250.  These  are  involutes,  not  of  the  pitch  circles,  but  of  the  base 
circles  EHY,  FGQ,  the  ratio  of  whose  radii  CE,  DF,  by  reason  of 
the  similar  triangles  ACE,  ADF,  is  the  same  as  that  of  AC,  AD,  the 
radii  of  the  pitch  circles.    Being  simultaneously  generated  during  the 
rotation  by  a  point  which  lies  always  in  the  common  tangent  to  the 
base  circles,  that  is  to  say,  in  the  common  normal  to  the  involutes, 
these  curves  will  be  tangent  to  each  other  throughout  the  generation  : 
and  since  this  common  normal  always  cuts  CD  at  the  same  point  A, 
they  may  be  used  as  outlines  of  teeth  for  the  wheels  to  which  they 
respectively  belong.     Thus  FI,  similar  to  GE,  will  drive  FH,  which 
is  similar  to  KE,  in  the  direction  indicated  by  the  arrow,  with  a  con- 
stant velocity  ratio  ;  the  locus  of  contact  being  the  line  FE.     The 
two  curves  are  shown  in  the  diagram  in  one  intermediate  position,  so 
that  the  action  will  be  readily  traced  without  farther  explanation  ; 
and  it  is  evident  that  this  action  is  precisely  equivalent  to  the  rolling 
together  of  the  pitch  circles  LM,  RS. 

251.  Now,  since  the  involute  does  not  extend  within  its  base  cir- 
cle, the  curves  shown  in  Fig.  148  are  of  the  greatest  possible  length  ; 
supposing  the  lower  wheel  to  drive,  the  action  begins  at  F  and  ends 
at  E.     Both  curves  are  continuous,  and  there  is  no  division  into  face 
and  flank  ;  but  it  is  evident  that  the  points  x  and  z  will  meet  at  A, 
the  involutes  then  occupying  the  positions  OAU,  PAW:  also  that 
the  ratio  of  the  arc  Az  to  the  arc  FO,  is  the  same  as  that  between  the 
arcs  Ax,  HP. 

•  Seasoning,  in  like  manner,  as  to  the  arcs  of  recess,  and  recollecting 
that  the  arcs  FO,  HP,  are  each  equal  to  FA,  while  OG,  PE,  are 
each  equal  to  AE,  we  have 

Approach  _  FA  __  CE  __  AC 
Eecess      ~  Ztf  ~~  ~DF  ~  AD  ' 


That  is  to  say,  if  each  involute  be  long  enough  to  extend  to  the  root 
of  the  other,  the  arcs  of  approach  and  recess  will  be  to  each  other  in 
the  direct  ratio  of  the  radii  of  the  base  circles,  or  pitch  circles,  of  the 
driver  and  the  follower  respectively. 

In  these  circumstances,  the  smaller  of  two  wheels  must  drive  in 
order  to  secure  more  receding  than  approaching  action.  But  as  it  is 
not  necessary  to  make  the  curves  so  long,  we  can  vary  the  proportion 


APPROACH   AND    KECESS. 


145 


between  the  arcs  of  approach,  and  recess  at  pleasure,  by  properly  regu- 
lating the  heights  of  the  teeth,  whatever  the  relative  diameters  of  the 
wheels. 

252.  Thus  in  Fig.  149,  which  represents  teeth  of  practical  propor- 
tions, one  pair  is  shown  in  contact  at  A,  another  pair  as  just  quitting 


FIG.  149. 

contact  at  B,  the  lower  wheel  driving  as  indicated  by  the  arrow.  The 
receding  action,  then,  continues  while  the  point  of  contact  travels 
from  A  to  B,  a  distance  equal  to  01  measured  on  the  base  circle. 
Meantime  the  linear  motion  of  the  driver,  measured  on  the  pitch  cir- 
cle, will  be  A  G,  which  is  greater  than  01  in  the  proportion  of  the 
radius  of  the  pitch  circle  to  that  of  the  base  circle  ;  but  since  those 
two  arcs  measure  the  same  angle,  either  is  readily  found  if  the  other 
be  given. 

Supposing,  then,  that  the  angle  of  recess  is  assigned  ;  we  first  ascer- 
tain the  length  of  the  arc  01  on  the  base  circle  by  which  it  is  meas- 
ured, and  set  off  AB  on  the  line  of  action,  equal  to  it ;  then  the  tops 
of  the  driver's  teeth  are  limited  by  a  circle  through  B,  about  that 
wheel's  centre. 

A  radius  through  B  cuts  the  pitch  circle  in  H\  if  GH  be  equal  to 
half  the  thickness  of  the  tooth  as  determined  by  the  pitch,  the  con- 
struction is  just  possible,  and  the  tooth  will  be  pointed.  If  GH  be 
greater  than  that,  the  conditions  are  impracticable  ;  but  if  it  be  less, 
the  tooth,  as  in  the  figure,  will  be  of  sensible  breadth  at  the  top. 

By  a  similar  process  the  height  of  the  follower's  tooth  for  any  as- 
signed angle  of  approach  is  determined,  and  the  pitch  circles  having 
been  subdivided  as  usual  into  equal  parts,  the  construction  is  com- 
pleted by  drawing  the  reverse  involutes  for  the  backs  of  the  teeth, 
and  providing  suitable  clearing  spaces.  These  may  be  required  to  ex- 
tend some  distance  within  the  base  circles,  and  if  so  they  may  be 
10 


146  ORIGINAL  PITCH   CIRCLE. 

bounded  as  in  the  figure  by  radial  lines  tangent  to  the  involutes  at 
their  roots. 

253.  Peculiar  Properties  of  Involute  Teeth. — In  the  operations  above 
described,  the  line  of  action  FE  was  drawn  at  pleasure,  and  the  ar- 
gument in  no  wise  depends  upon  its  obliquity.     Consequently,  for  a 
given  pair  of  pitch  circles  an  infinite  number  of  pairs  of  base  circles 
may  be  assigned,  and  the  converse  is  evidently  true,  since  the  common 
tangent  of  any  two  given  base  circles  will  always  cut  the  line  of  cen- 
tres, be  the  same  greater  or  less,  into  segments  having  the  same  ratio 
as  their  radii.     From  which  follow  two  important  practical  deduc- 
tions, viz.: 

1.  Any  two  wheels  with  involute  teeth,  of  which  the  pitch  arcs  on 

the  base  circles  are  equal,  will  gear  correctly  with  each  other. 

2.  The  velocity  ratio  will  not  be  affected  by  any  change  in  the  dis- 

tance between  their  centres. 

"We  have  seen  that  wheels  with  epicycloidal  teeth  may  be  made  in- 
terchangeable ;  but  this  second  peculiarity  gives  the  involute  tooth  an 
advantage  over  every  other,  of  special  importance  in  mechanism  re- 
quiring the  greatest  smoothness  and  uniformity  of  action.  The  veloc- 
ity ratio  will  be  correct,  although  the  wheels  be  improperly  located  at 
the  outset,  and  will  remain  so  in  spite  of  wear  in  the  bearings  ;  also, 
the  backlash  may  at  any  time  be  reduced  to  a  minimum,  by  bringing  the 
axes  as  close  together  as  they  can  be  without  causing  the  teeth  to  bind. 

On  the  other  hand,  the  obliquity  is  constant,  and  in  general  greater 
than  the  mean  obliquity  for  epicycloidal  teeth  having  the  same  angle  of 
action.  While  this  may  be  a  serious  objection  to  the  involute  form  for 
heavy  work,  it  may  also  be  a  positive  advantage  in  light  mechanism  where 
smoothness  of  action  is  all-important.  For  the  side  pressure  will  al- 
ways keep  the  axes  at  the  greatest  possible  distance  from  each  other, 
which  tends  to  prevent  shaking  if  there  be  any  looseness  in  the 
bearings. 

254.  Original  Pitch  Circle. — Of  the  infinite  number  of  pitch  circles 
which  may  be  assigned  for  any  given  wheel,  there  is  evidently  but  one 
upon  which  the  tooth  and  space,  as  in  Fig.  149,  are  measured  by  equal 
arcs.     This  is  appropriately  called  the  original  pitch  circle,  being  the 
one  given  or  assumed  in  laying  out  a  pair  of  wheels  with  teeth  and 
spaces  equal  as  above  described,  and  it  determines  what  may  be  called 
the  proper  obliquity  for  both  wheels. 

Suppose  now  that  a  wheel  has  been  made,  the  teeth  being  involutes 
of  a  given  base  circle,  the  pitch  circle  unknown  ;  and  let  it  be  re- 
quired to  construct  another  which  shall  gear  with  it  in  such  wise  that 
the  teeth  and  spaces  on  the  pitch  circles  shall  be  equal  as  in  the  fig- 


RACK   AND   WHEEL.  .  147 

ure.  In  order  to  do  this,  it  is  necessary  first  to  find  the  proper  obliq- 
uity of  the  given  wheel.  This  may  be  done  graphically  as  follows  : 
draw  two  radii,  the  first  bisecting  a  tooth,  the  second  bisecting  an  ad- 
jacent space,  of  the  given  wheel.  Then  bisect  the  angle  between  these 
lines  by  another  radius  ;  this  will  cut  the  involute  outline  of  the  tooth 
in  a  point  through  which  the  original  pitch  circle  must  pass.  This 
third  radius  being  taken  as  the  line  of  centres,  the  tangent  to  the  base 
circle,  through  the  point  last  mentioned,  will  be  the  line  of  action 
having  the  required  obliquity. 

255.  Considerations  Affecting  the  Obliquity.  —  The  obliquity  is,  as 
above  stated,  abstractly  arbitrary.  But  it  will  presently  appear  that 
under  some  circumstances  there  are  definite  relations  between  the  ob- 
liquity, the  velocity  ratio  and  the  arc  of  recess,  such  that  if  either 
two  be  assigned,  the  other,  in  many  cases,  can  vary  only  within  a  quite 
narrow  range.  For  practical  guidance  when  not  trammeled  by  the 
considerations  alluded  to,  it  may  be  stated  that  experience  has  shown 
that  for  ordinary  purposes  the  obliquity  should  not  exceed  from  15° 
to  17°.  When  it  is  no  greater  than  this,  it  is  safe  to  say  that  even  for 
heavy  work  these  teeth  are  in  every  respect  equal  to  the  epicycloidal 
or  any  others  ;  and  in  addition  to  the  advantages  previously  men- 
tioned, it  will  be  noted  that  their  form  is  essentially  a  strong  one, 
spreading  out  rapidly  toward  the  base. 

25G.  Rack  and  Wheel  with  Involute  Teeth.—  In  Fig.  150,  LM  being 
the  pitch  circle,  and  TT  the  pitch  line  of  the  rack,  the  base  circle  is 
determined  as  in  Fig.  148,  by  drawing 
FE  at  an  arbitrary  angle  through  A, 
the  point  of  tangency,  and  letting  fall 
upon  it  the  perpendicular  CE.     Then 
while  a  marking  point  travels  from  E 
to  F,  with  a  linear  velocity  equal  to 
that  of  the  circumference   EHY,  it. 
will  trace  upon  the  plane  of  the  wheel 
the  involute  FH.     The  rack  meantime 
moves  to  the  left  side  with  a  linear  ve- 
locity equal  to  that  of  the  pitch  circle, 
which  is  greater  than  that  of  the  base  circle  in  the  ratio  of  A  C  to  CE. 

Consequently,  when  the  pencil  reaches  F  the  point  E  of  the  rack 
will  have  moved  to  /,  and  we  shall  have  EF  perpendicular  to  EC,  El 
perpendicular  to  A  C,  and  also 


iL          • 

EF  ~:  EC  ' 


148 


INSIDE   GEARING. 


M 


pn  „._..._.... 


i 

FIG.  151. 


therefore  IF,  traced  by  the  pencil  upon  plane  of  the  rack,  will  be  a 
right  line  perpendicular  to  EF. 

Drawing  EK  similar  to  HF,  and  EG  similar  to  IF,  and  supposing 
the  pinion  to  drive,  the  action  begins  at  E,  ending  at  F;  and  as  in 
Fig.  149,  the  relative  amounts  of  approaching  and  receding  action 

may  be  varied  at  pleasure  by  alter- 
ing the  heights  of  the  teeth.  In 
this  diagram,  as  in  Fig.  148,  prac- 
tical proportions  are  disregarded 
for  the  sake  of  perspicuity  ;  the 
T  general  appearance  of  the  combi- 
nation under  reasonable  conditions 
is  shown  in  Fig.  151,  the  obliquity 
being  20°. 

The  less  the  obliquity,  the  shorter  will  be  the  acting  face  of  the  rack 
for  a  given  angle  of  action,  and  the  less  will  be  the  amount  of  ap- 
proaching action  attainable  ;  when  the  obliquity  is  nil,  the  rack  tooth 
degenerates  into  a  point,  and  the  wheel-tooth  becomes  an  involute  of 
the  pitch  circle,  as  illustrated  in  Fig.  121.  The  action  is  wholly  re- 
ceding, but  the  sliding  is  excessive  and  the  wear  confined  to  one  point 
on  the  rack,  thus  more  than  counterbalancing  all  advantages  unless 
the  pressure  is  very  moderate. 

257,  Annular  Wheels  with  Involute   Teeth. — The  construction,  as 
shown  in  Fig.  152,  is  substantially  the  same  as  in  the  case  of  outside 
gearing.      And    this  form  of 

tooth  would  appear  to  be  ex- 
tremely well  adapted  for  inside 
gearing  requiring  unusual 
smoothness  of  action,  to  which, 
as  above  suggested,  the  con- 
stant obliquity  is  favorable, 
while,  owing  to  the  small  dif- 
ference in  the  lengths  of  the 
acting  curves,  there  is  very 
little  sliding.  This  diagram  is 
so  similar  to  Fig.  150,  that  no 
explanation  is  necessary,  be- 
yond calling  attention  to  this  FIG.  153. 
fact,  viz.:  since  the  action  must  begin  or  end  at  E,  the  root  of  the 
pinion's  teeth,  the  greatest  possible  height  of  the  wheel-tooth  is  de- 
termined by  drawing  a  circle  about  D  through  that  point. 

258,  Low-numbered  Pinions  with  Involute  Teeth. — If  two  wheels  be 


LOW-NUMBERED   PINIONS. 


149 


of  equal  size,  the  least  number  of  teeth  which  can  be  given  to  each 
depends  upon  the  obliquity. 
For  if  this  be  assigned,  the 
points  F  and  E,  Fig.  153,  are 
thereby  determined ;  and  the 
arcs  FG,  EH,  on  the  base  cir- 
cles, are  each  equal  to  EF. 
Taking  these  arcs,  then,  as  the 
measures  of  the  pitch  angles, 
and  making  the  teeth  and 
spaces  on  the  pitch  circles 
equal,  the  combination,  as  rep- 
resented in  the  figure,  is  just  a 
limiting  case,  and  the  number 
of  teeth  the  least  possible. 

If  the  wheels  are  to  be  com- 
plete ones,  a  fractional  tooth 
being  impossible,  the  arc  FG 
must  be  contained  an  exact 
number  of  times  in  the  cir- 
cumference, or,  if  there  be  a 
remainder,  the  next  higher  integer  must  be  taken  as  the  least  number 
of  teeth.  In  the  latter  event  the  arrangement  will  be  practically  ser- 
viceable, for  though  the  angles  of  approach  and  recess  will  still  be 
equal,  their  sum  will  be  greater  than  the  pitch.  Each  of  these  angles 
is  in  point  of  fact  a  little  greater  than  the  obliquity,  but  the  excess,  as 
can  easily  be  verified  by  calculation,  is  so  small  that  the  least  number 
of  teeth  for  practical  purposes  is  at  once  correctly  ascertained,  by  divid- 
ing the  circumference  by  twice  the  given  obliquity  :  and  thus  for  equal 
wheels  we  have 


FIG.  153. 


OBLIQUITY. 

10° 


15° 

18° 
20° 
30° 


LEAST  No.  OF  TEETH. 
18 

15 

12 

.   10 


G. 


259.  Supposing  the  lower  wheel  in  Fig.  153  to  drive,  then  the 
follower  may  be  equal  to  it,  but  cannot  be  less.  It  may,  however,  be 
made  as  much  greater  as  we  please  ;  —  a  moment's  study  of  the  diagram 
will  show  that  this  same  wheel  can  drive  even  the  rack  shown  in 
dotted  outlines. 


150 


LOW-NUMBERED   PINIONS. 


A  sufficient  increase  in  the  diameter  of  the  follower  will  permit 
the  tooth  GE  of  the  driver  to  be  lengthened  to  an  extent  limited  by 
its  intersection  at  P  with  the  reverse  involute,  forming  the  back  of 
the  tooth.  About  D,  describe  a  circle  through  P9  cutting  the  line  of 
action  in  B  ;  draw  J5/  parallel  to  EC,  cutting  the  line  of  centres  in  /: 
then  if  the  driver's  tooth  be  pointed,  BI  will  be  the  least  radius  of  the 
follower  which  will  secure  the  greatest  possible  amount  of  receding 
action,  viz.  :  an  arc  on  the  base  circle  equal  in  length  to  AB. 

260.  This  leads  directly  to  the  consideration  that  the  pitch  of  the 
driver  may  be  made  greater  than  in  the  figure,  without  changing  the 

obliquity  ;  for  by  increasing  the 
angle  FDP,  the  point  of  the 
leading  tooth  may  be  made  to 
fall  upon  the  line  of  action,  the 
tooth  and  space,  as  measured  on 
the  pitch  circle,  still  remaining 
equal  to  each  other. 

This  is  illustrated  in  Fig. 
154,  FG  being  the  pitch  on  the 
base  circle,  and  LM,  the  corre- 
sponding arc  on  the  pitch  cir- 
cle P,  on  the  line  of  action, 
being  the  point  of  the  tooth, 
the  involute.  PK  must  bisect 
LM  in  N\  then,  PD  will  bi- 
sect MN  in  /,  and  HD,  which 
bisects  FK,  will  also  bisect 
LN  in  H.  The  obliquity  being 
given,  the  angle  FDL,  which  is 
equal  to  MDG,  may  be  calcu- 
lated ; 

then  letting  FDL  =  a, 
FDH  =  x 


FIG.  IM. 


"we  shall  have 


i  Pitch  =  LDH=x  +  a, 

FDP  =  3(x  +  a)  -  a  =  3x  +  2«, 
FDG  =  LDM  =  4(a?  +  a). 

But  FP,  the  trigonometrical  tangent  of  the  arc  FI,  is  equal  to  the 
arc  FG  ;  that  is, 

tan  (3x  +  2a)  =  arc  (4z  +  40). 


MAXIMUM   PITCH   FOR  GIVEN   OBLIQUITY.  151 

This  equation  cannot,  we  believe,  be  solved  by  any  process  less 
laborious  than  the  tentative  one  of  assuming  a  value  for  x,  and  if  it 
fail  to  satisfy  the  equation,  increasing  or  diminishing  it,  as  the  case 
may  be,  until  it  does.  Nor  does  the  situation  afford  any  key  to  a 
graphic  construction,  since  the  distance  to  be  set  off  on  the  line  of 
action,  and  the  angle  subtended  by  an  arc  equal  to  it  on  the  circum- 
ference, are  both  unknown. 

This,  however,  is  of  little  consequence,  as,  if  the  obliquity  be  as- 
signed, there  can  be  no  certainty  that  the  pitch  arc  when  found  would 
be  an  aliquot  part  of  the  circumference  :  and  it  is  of  much  more  prac- 
tical interest  to  ascertain  the  result  of  assigning  a  specific  number  of 
teeth,  or  in  other  words  a  definite  pitch. 

261.  When  this  is  done  the  case  becomes  more  manageable.     Consid- 
ering first  the  graphic  process  :  The  radius  FD  of  the  base  circle  and 
the  pitch  angle  FDO  being  given,  we  have  first  simply  to  rectify  the 
arc  FG  and  set  off  FP  equal  to  it,  in  a  direction  perpendicular  to 
DF.     Drawing  the  involutes  PG  and  PK,  the  tooth  is  completed  as 
to  its  acting  outlines  ;  now  bisect  the  angle  FDK  by  the  radial  line 
DH,  draw  PD,  and  bisect  HDP  by  DN:  this  latter  line  will  cut  the 
involute  PK  in  the  point  N,  through  which  is  drawn  the  original 
pitch  circle  cutting  FP  in  A,  which  locates  DA  the  line  of  centres 
and  determines  the  proper  obliquity. 

The  pinion  thus  formed  is  barely  capable  of  driving,  the  whole  path 
of  contact  FP  being  just  equal  to  FG,  the  pitch  arc  on  the  base  cir- 
cle :  and  the  minimum  radius  of  the  follower  will  be  determined  by 
drawing  PC  perpendicular  to  the  line  of  action,  cutting  DA  produced 
in  C  the  centre. 

262.  Otherwise,  by  computation  :  Knowing  the  radius  FD  and  the 
length  of  FP,  the  angle  FDPmay  be  found,  whence,  the  angle  FDG 
being  given,  the  angles  GDK,  FDK,  may  also  be  found,  and  we  shall 
then  have 

FDL  =  J  (GDK  -  FDK}. 

Now,  FA,  the  trigonometrical  tangent  of  the  angle  ADF,  is  equal 
to  the  arc  of  the  base  circle,  which  measures  the  angle  ADL  ;  there- 
fore, let 

ADF  =  y, 

FDL  =  a, 

and  we  have 

tan  y  =  arc*  (y  +  a). 

This  equation  may  be  solved  by  the  tentative  process  above  de- 


152 


GREATEST  PATH  OF  COXTACT. 


scribed  ;  and  the  angle  ADF  thus  determined  is  equal  to  the  proper 
obliquity.  Knowing  this  angle  and  the  side  FD,  we  can  now  find 
FA,  equal  to  the  arc  of  approach  on  the  base  circle,  which  taken  from 
FP  gives  AP  equal  to  the  arc  of  recess,  and  the  triangles  ADF,  A  CP, 
being  similar,  the  radius  PCis  readily  ascertained. 

263.  In  Fig.  154  the  pitch  arc  FG  is  72°,  so  that  the  diagram  rep- 
resents the  construction  of  a  five-leaved  pinion  just  capable  of  driving. 
By  comparing  this  figure  with  the  preceding  one,  it  will  be  seen  that 
when  the  point  of  the  tooth  thus  falls  upon  the  line  of  action,  we  have 
the  maximum  pitch  for  a  given  obliquity,  or  the  minimum  obliquity 
for  a  given  pitch. 

In  this  case,  we  find  by  computation  in  the  manner  above  explained, 
the  following  values,  viz. : 

Obliquity 28°  8' 

Arc  of  Recess 4  pitch. 

Least  No.  for  Follower 6.74. 

Practically,  then,  a  pinion  of  five  leaves  is  capable  of  working  with 

another  of  seven,  but  no  less 
without  exceeding  the  obliquity 
above  given.  By  increasing  the 
obliquity,  the  number  for  the 
follower  may  be  reduced,  but  it 
will  subsequently  be  shown  that 
five  is  not  only  the  least  number 
for  equal  wheels,  but  also  the 
least  that  can  either  drive  or  be 
driven  at  all., 

264.  The  greatest  possible 
path  of  contact  is  the  distance 
between  the  feet  of  the  perpen- 
diculars let  fall  from  the  centres 
of  the  wheels  upon  the  line  of 
action.  It  has  been  pointed  out 
that  if  the  obliquity  be  assumed, 
there  is  no  certainty  that  this 
distance  will  be  an  aliquot  part 
of  the  circumference  of  the  base 
circle.  But  if  the  numbers  of 
teeth  be  given  for  a  proposed 
FIG.  155.  pair  of  wheels,  it  may  be  made 

so,  and  the  obliquity  thus  determined. 


INVOLUTE   WITH   EPICYCLOIDAL  EXTENSION.  153 

For  example,  in  Fig.  155,  let  the  numbers  be  six  for  the  upper 
wheel  and  five  for  the  lower.  About  any  centre  C,  with  a  radius 
measuring  6  on  any  scale  of  equal  parts,  describe  a  circle  ;  perpendicu- 
lar to  any  radius  CE,  set  off  EF  equal  to  the  arc  EH  which  measures 
the  pitch  ;  perpendicular  to  EF,  and  on  the  side  opposite  EC,  set  off 
FD  measuring  5  parts  on  the  same  scale,  and  draw  CD,  cutting  FE 
in  A.  Then  taking  AC,  AD,  as  the  radii  of  the  pitch  circles,  and 
CE,  DF,  as  the  radii  of  the  base  circles,  the  path  of  contact,  EF, 
which  is  equal  to  one-sixth  of  the  circumference  of  the  upper  base 
circle,  will  measure  one-fifth  of  that  of  the  lower  one. 

And  this  construction  fixes  the  minimum  obliquity  for  the  given 
numbers  of  teeth  ;  it  being  evident  that  a  decrease  of  this  obliquity 
will  at  once  shorten  the  path  of  contact  and  enlarge  the  base  circles. 

265.  Involute  Tooth  with  Epicycloidal  Extension. — The  path  of  con- 
tact, as  above  determined,  is  exactly  equal  to  the  pitch  ;  now  in  prac- 
tice the  arc  of  action  must  always  be  greater  than  the  pitch,  and  yet 
it  may  be  desirable  to  use  involute  teeth  in  circumstances  like  those 
represented  in  Fig.  155.  The  involute  does  not  continue  within  its 
base  circle,  but  the  face  HF  of  the  blunted  tooth  may  be  extended  as 
shown  in  dotted  lines,  thq  curve  beyond  F  being  an  epicycloid  gener- 
ated by  a  circle  whose  diameter  is  AD,  which  will  work  correctly  with 
the  radial  flank  bounding  the  clearing  space  in  the  lower  wheel.  A 
similar  construction  is  applicable,  of  course,  to  the  other  wheel,  if  its 
tooth  be  originally  so  much  blunted  as  to  make  it  desirable  ;  and  in 
.this  way  a  sufficient  angle  of  action  may  be  secured,  and  the  involute 
tooth  employed  in  cases  where  it  would  otherwise  be  inadmissible. 


Limiting  Numbers  of  Teeth  for  Given  Arcs  of  Recess. 

266.  In  the  employment  of  the  involute  system,  if  a  definite  pitch 
and  arc  of  recess  be  assigned  for  either  wheel  of  an  engaging  pair,  there 
will  be  a  limit  to  the  number  of  teeth  which  can  be  given  to  the  other, 
as  was  shown  to  be  the  case  with  the  epicycloidal  tooth. 

Practically,  a  knowledge  of  these  limiting  numbers  is  just  as  neces- 
sary or  convenient  in  using  one  form  of  tooth  as  it  is  in  using  another. 
Next  to  the  epicycloid,  the  involute  is  the  curve  most  extensively  used, 
and  in  the  mode  of  generation,  as  well  as  in  the  peculiarities  of  the 
action,  there  is  a  marked  contrast  between  the  two,  and  for  these  rea- 
sons a  special  interest  attaches  to  a  comparison  of  the  limiting  num- 
bers for  these  forms  under  like  conditions.  We  shall  therefore  pro- 
ceed to  investigate  a  method  of  determining  these  limits  for  the  same 


154 


LIMITING   NUMBERS   OF  TEETH. 


values  of  the  arc  of  recess  which  were  assumed  in  the  tables  for  epi- 
cycloidal  teeth,  viz. : 

1.  Kecess  =  Pitch.     2.  Recess  =  |-  Pitch.     3.  Recess  =  f  Pitch. 

267.  In  Fig.  156,  let  C,  D  be  the  centres  of  two  wheels,  VW  and 
HY  arcs  of  the  pitch  circles  tangent  at  A,  FQ  the  line  of  action, 
CQ  and  DFihe  radii  of  the  base  circles,  and  let  the  lower  wheel  drive 

in  the  direction  indicated 
by  the  arrow. 

The  proportions  are  such 
that  the  tooth  of  the  driver 
is  pointed,  and  that  its 
point,  Q,  extends  just  to 
the  root  of  the  other  invo- 
lute ;  also,  the  breadth  MN 
of  the  tooth  being  equal  to 
the  space  NL,  both  meas- 
ured on  the  pitch  circle, 
there  is  no  backlash. 

This,  evidently,  is  just  a 
limiting  case ;  the  pitch 
ML  and  the  arc  of  recess 
A M  being  assigned,  the  in- 
volutes bounding  the  tooth 
GQK  must  pass  through 
the  points  M  and  N,  what- 
ever be  the  radius  of  the 
base  circle,  and  the  obliq- 
uity TAF9  which  causes 
their  intersection  to  fall  on 
the  line  of  action,  is  clearly 
the  greatest  possible,  which 

as  clearly  gives  the  least  possible  value  for  A  C. 

The  diagram,  then,  represents  the  problem  as  already  solved,  and 

it  remains  to  deduce  from  it  the  means  of  effecting  the  solution  in 

any  given  case. 

268.  Draw  QD,  which  cuts  the  lower  base  circle  in  0 ;  then  the 
path  of  contact  FQ  is  equal  to  the  arc  FG,  and  is  also  the  trigonomet- 
rical tangent  of  the  arc  FO ;  that  is  to  say,  we  have,  as  an  equation 
of  condition, 

tan  FO  =  arc  FG. 


FIG.  156. 


METHOD   OF   COMPUTATION.  155 

The  line  of  centres  cuts  the  lower  base  circle  in  B  ;  through  L,  A, 
and  N,  draw  LR,  A  I,  NH,  involutes  of  the  lower  base  circle,  and 
through  A  draw  also  A  J,  an  involute  of  the  upper  base  circle. 

Then  the  arc  QJ  is  equal  to  QA,  the  trigonometrical  tangent  of 
the  angle  QCA  ;  the  arc  FI  is  equal  to  FA,  the  trigonometrical  tan- 
gent of  the  angle  FDA  :  and  the  angles  QCA,  FDA,  are  equal  to 
each  other  and  to  the  obliquity  TAF. 

We  shall  now  have 

Fa  =  FR  +  2RH  ...........................  (1) 

FO  =  FG  -  OG 

OG  = 
(  KH= 
\MN=NL, 

RG  =  2£I  +  RH 


B  ,  =  2BI,  .'  ........         KG  =  2BI+  HG 


FO  =  FR  +  2RH  -  BI-—~ 


or  FO  =  FR  +      —  -  BI 

a 


Also  FI   =  FR  +  #/=  FB  +  BI 


269.  Now  when  the  definite  values  mentioned  in  (266)  are  assigned 
for  the  arc  of  recess,  these  equations  will  be  modified  as  follows  : 

I.  RECESS  =  PITCH. 
In  this  case  we  have 

RI  =  0,      ) 

RH  =  IH,  [•  whence 

FR  =  FI,] 

FG  =  FI  +  2IH.  .............  1. 


......... 

FI=  FB  +  BI  ........  3. 

And  for  a  pinion  driving  a  rack, 

»  FR  =  FI  =  0,  ) 

FB  =  0,  [•  whence 
BI  =  0,  ) 


156                                     LIMITING   NUMBERS   OF   TEETH. 
FG  =  2IH 1. 

FO  =  ^j~- 2. 

II.  RECESS  =  f  PITCH. 
We  now  have 

r>  T         HrL  r>  rr o    T>  T 

Ml   =  — g- ,  Or  Kli   =  A  £lJL) 

which  gives 


FG  =  FR  +  47?7  ........   1.  )  vr  _    VT  , 

FI  =  FR  +  RI  .........   3.f    •   •  FG~ 

and 


+  Mr-BI....  2.)      .. 
=  FB    +131....  3.  ) 


FR+  RI 
When  the  pinion  drives  a  rack, 

FR  =  0,     )          j  FG  =  477, 


)          (FG  =  4FI, 
,  J   '      1  FO  =  FB  +  2^7. 

III.  RECESS  =  §  PITCH. 


With  this  value 

RI  =  %RH,  or  EH  =    ~. 
And  therefore 


Also 


FR  +  RI  =  FB  +  El. . . .  3.  ) 
And  in  the  case  of  a  pinion  driving  a  rack, 


+   SRI.  ...     1.  )  TTT/^  _.    7:77- 

FI  =FR  +  RI 3.      '  •  ^  "  FI 


~. 

270.  How  suppose  the  pitch  and  the  arc  of  recess  to  be  assigned  for 
the  driver.     The  arcs  IH  and  HO  are  then  known  ;  and  if  we  assume 


BACK   DRIVING — SPECIAL  CONSTRUCTION. 


157 


a  value  for  FI,  we  can  find  the  corresponding  values  of  BI  and  FB, 
which  combined  with  the  others  will  give  certain  valuer  for  FO  and 
FG  ;  and  these  are  finally  to  be  tested  to  see  whether  they  will  satisfy 
the  equation  of  condition, 

tan  FO  =  arc  FG. 

Having  at  last  found  in  this  way  the  obliquity  which  under  the 
assigned  conditions  makes  FQ  equal  in  length  to  the  arc  FG9  we 
know  also  the  length  of  FA  \  whence  A  C,  the  minimum  radius  of 
the  follower,  is  readily  ascertained. 

If,  on  the  other  hand,  the  wheel  whose  pitch  and  arc  of  recess  are 
given  be  the  follower ;  we  then  know  the  values  of  QA  and  the  arc 
QJ,  which  are  equal  to  each  other,  and  can  thence  find  the  angle 
ACQ,  equal  to  ADF  and  to  the  obliquity.  We  then,  knowing  the 
arcs  FB,  FI,  assume  a  value  for  RI,  and  proceed  as  before  to  see 
whether  the  equation  of  condition  is  satisfied. 

271.  When  the  given  driver  becomes  a  rack,  the  least  radius  for 
the  follower  is  most  readily  ascertained  by  a  special  construction, 
which  is  given  in  Fig.  157. 

We  have  here  an  exact 
limiting  case,  the  rack 
tooth  being  pointed,  and 
the  whole  path  of  contact 
just  equal  to  the  pitch  arc 
on  the  base  circle ;  thus, 
the  rack  driving  in  the 
direction  of  the  arrow,  a 
marking  point  in  going 
from  Q  to  F,  generates  the 
right  line  $£7  and  the  in- 
volute QG,  the  latter  cut-  FIG.  157. 
ting  the  pitch  circle  at  M.  The  action,  then,  begins  at  Q,  and  ends 
at  F,  the  root  of  the  involute  FS9  which  latter  curve  cuts  the  pitch 
circb  in  L.  Since  the  tooth  and  space  are  to  be  equal,  the  side  XK 
of  the  pinion's  tooth  must  bisect  the  arc  ML  at  N  \  and  it  must  also 
be  tangent  to  the  side  FV  of  the  rack-tooth,  at  P  its  intersection  with 
the  other  line  of  action  PAZ,  to  which  FVis  perpendicular. 

It  will  now  be  seen  that  while  the  marking  point  is  moving  from  Q 
to  A,  it  generates  the  curve  QM,  the  pinion  rotating  through  the  arc 
MA  ;  in  the  reverse  motion,  a  marking  point,  in  going  from  P  to  A, 
traces  the  curve  PN,  while  the  pinion  turns  through  the  arc  NA. 


158  BACK  DRIVING  —  SPECIAL  CONSTRUCTION. 

But  we  have 

MA  +  AN  =  MN  =  i  ML, 
whence  QA   +  PA  =  i  QF, 

or  PA  ==  i  QF  -  QA  ; 

also,  QA  =  QF—  AF; 

therefore,  since  QFis  equal  to  the  pitch  arc,  and  AFto  the  arc  of  re- 
cess (both  measured  on  the  base  circle),  PA  is  known  when  these  arcs 
are  given. 

We  have,  then,  this  simple  graphic  process  :  Draw  any  line  FQ,  and 
let  it  represent  the  pitch  ;  set  off  upon  it  a  distance  FA,  making 

FA  _  arc  of  recess  . 
~FQ         pitch  arc 

next  determine  PA  as  above,  and  construct  the  right-angled  triangle 
APF.  Then  the  angle  PAF  is  twice  the  obliquity,  and  A  T  bisecting 
it  is  the  pitch  line  of  the  rack.  Draw  A  C  perpendicular  to  A  T,  and 
FC  perpendicular  to  FQ  ;  these  will  intersect  at  C  the  centre  of  the 
pinion,  determining  CF  the  radius  of  the  base  circle  and  CA  that  of 
the  pitch  circle. 

272.  Now  introducing  the  same  definite  values  as  before  for  the  arc 
of  recess,  we  shall  have, 

I.  EEC  ESS  =  PITCH. 


II.  RECESS  =  J  PITCH. 


III.  RECESS  =  f  PITCH. 


In  these  oases,  then,  no  tentative  proceeding  is  involved  in  deter- 
mining the  least  number  of  teeth  for  the  pinion,  which  requires  merely 
the  solution  of  the  triangles  APF,  ACF. 

273.  In  Fig.  157  the  pinion's  tooth  has  a  sensible  breadth,  QX,  at 
the  top,  and  the  involutes  might  therefore  be  continued  as  shown  in 


RACK   DRIVING — LIMITING  CA 


dotted  lines,  and  the  approaching  action  thereby  prolonged. 

smaller  the  assigned  arc  of  recess,  the  less  is  AP,  and  the  narrower 

will  be  the  tooth  at  the  top,  when  as  in  the  figure  the  total  arc  of  ac- 


m         \ 


tion    is  made   equal   to   the  ^  ,  -    ; 
pitch:  there  will,   then,   be  /      w 
some  value  of  the  arc  of  re-  y 
cess    which    will    cause  the 
points  Q  and  Xto  coincide. 

And  it  is  apparent  that 
when  the  limit  is  reached,  as 
in  Fig.  158,  the  pinion  will 
have  the  least  possible  num-. 
ber  of  teeth  which  can  be 
driven  by  a  rack. 

Now  if  in  this  diagram  the 
arc  FI  be  given,  we  know 
FA,  the  trigonometrical  tangent  of  the  angle  ACF,  which  is  always 
one  half  the  angle  PAF:  whence  AP  maybe  found.  We  have 
also 


FIG.  iss. 


and 


AQ  =  AF  -  ZAP, 

AQ  +  AF=  FQ  = 


•     •     (A) 


Again,  in  the  equations  deduced  in  (268),  viz. : 

FG  =  FR  +  2  RH, 

FO  =  FR  +  3  ^H  -  BI, 

•     2 

we  have 
whence 

and 
or 


FG  =  2  RH, 

FO  =  *J¥*  -  BI, 

FO  =     FG  -  BI. 


(B) 


274.  From  these  data  we  may  now  solve  the  problem  in  the  tenta- 
tive manner  before  described. 

Taking  any  convenient  value  for  the  radius  FC,  we  first  assume  the 
arc  FI,  and  find  the  corresponding  values  of  FA,  of  the  angle  A  CF, 
and  of  the  arc  BI.  Then  solving  the  triangle  APF,  we  determine 
AP,  whence  we  find  FQ,  as  in  Eq.  (A). 

The  length  of  FQ  being  equal  to  that  of  the  arc  FG,  we  next  find 
the  length  of  the  arc  FO  as  in  Eq.  (B),  and  knowing  the  radius,  as- 
certain the  value  of  this  arc  in  degrees. 


160  LOW-NUMBERED   PINIONS. 

Finally,  we  apply  the  test  of  determining  the  length  of  the  trigono- 
metrical tangent  of  this  arc,  and  comparing  it  with  that  of  FQ  as  be- 
fore found.  If  these  two  values  are  equal,  the  assumed  value  of  FI 
is  correct ;  but  if  they  are  unequal  a  new  value  must  be  assumed,  and 
the  process  repeated  until  concordant  results  are  obtained. 

In  this  case  we  find  that  the  pitch,  at  the  limit  which  gives  a 
pointed  tooth  to  the  pinion,  corresponds  to 

No.  of  Teeth =  4.0906 

Obliquity =  41°  20'  45"  )  , 

Arc  of  Kecess =  -ff  Pitch,     f  Very ' 

275.  Now  referring  to  Fig.  153,  it  will  be  seen  that  in  constructing 
a  pair  of  equal  wheels,  making  the  pitch  arc  on  the  base  circle  equal 
to  the  greatest  possible  path  of  contact,  the  teeth  will  become  nar- 
rower at  the  top  as  the  obliquity  is  increased  ;  evidently,  then,  there 
is  a  limiting  value  of  the  obliquity  which  will  make  the  teeth  of  both 
wheels  pointed,  as  shown  in  Fig.  159. 

In  this  case  we  have  [Eq.  (B).  (273)], 


FO  =  1  FQ  -  BI-, 

and  also  FQ  =  2  FI,    .  • .    FQ  =  2  FA. 

Whence  FO  =  f  FI  -  BI-, 

but  BI  =  FI  -  FB,   .-.    FO  =  J  FI  +  FB. 

It  will  now  be  seen  that  by  assuming  a  value  for  FI,  all  the  quan- 
tities involved  in  the  above  expressions  may  be  determined  ;  and  the 
results,  as  before,  must  satisfy  the  equation  of  condition, 

tan  FO  =  arc  FG. 
By  this  process  we  find 

No.  of  Teeth =  4.6,256 

Obliquity. =  34°  10'  58.5" 

Arc  of  Eecess  (given) =  |  Pitch. 

If  complete  wheels  are  to  be  used,  then,  five  is  the  least  number  that 
can  drive  or  be  driven,  since,  as  shown  in  the  preceding  section,  four 
and  a  fraction  are  required  even  when  the  pinion  gears  with  a  pointed 
rack.  By  adopting  the  construction  shown  in  Fig.  155,  the  obliquity 


LIMITING    NUMBERS   OF   TEETH. 


161 


just  given  may  be  slightly  reduced,  and  we  shall  have  for  the  smallest 
possible  pair  of  complete  equal  wheels, 

No.  of  Teeth  ...........................   5 

Obliquity  ..............................   32°  8'  31" 

Arc  of  Eecess  ...........  ...............      Pitch. 


276.  The  tentative  processes  above  explained  are  undeniably  tedious, 
yet  when  systematically  conducted  they  are  less  so  than  might  be  sup- 
posed ;  and  in  the  preparation  of  the  following  tables  they  have  been 
continued  until,  with  a  radius 
of  10,  the  linear  values  of  the 
arc  FG  and  the  path  of  contact 
FQ  agreed  up  to  the  sixth  deci- 
mal place,  the  approximation 
being  therefore  to  within  the 
one-millionth  part  of  the  radius 
of  the  base  circle. 

It  was  not  to  be  expected  that 
for  a  given  number  of  teeth 
upon  either  wheel  the  radius  of 
the  other  as  thus  determined 
would  correspond  to  an  exact 
whole  number  of  teoth  upon  its 
periphery.  The  next  higher 
integer  being  of  course  taken 
as  the  tabular  number,  the 
tooth  would  not  then  be  abso- 
lutely a  pointed  one  ;  but  we 
have  not  made  any  correction 
for  the  slight  change  in  the  ob- 
liquity due  to  this  circumstance,  considering  its  value  as  appearing  in 
the  calculation,  which  is  given  to  the  nearest  second,  sufficiently  pre- 
cise for  all  practical  purposes. 

It  is  proper  to  point  out  that  these  tables  may  be  of  service,  not  only 
as  indicating  exact  limiting  cases,  which  are  to  be  avoided  when  pos- 
sible, but  upon  another  account,  viz.  :  although  in  general  terms  the 
obliquity  is  said  to  be  arbitrary,  yet  in  constructing  for  special  pur- 
poses wheels  with  a  given  arc  of  recess,  it  may  be  necessary  to  employ 
numbers  not  far  removed  from  limiting  values,  and  in  many  instances, 
it  will  be  seen,  the  possible  variation  in  the  obliquity  is  confined  to 
quite  a  narrow  range. 
11 


FIG.  159. 


162 


INVOLUTE   AND  EPICYCLOID   COMPARED. 

Limiting  Numbers  of  Teeth. 


EECESS  =  PITCH.                                     TOOTH  =     SPACE. 

INVOLUTE. 

EPICYCLOIDAL. 

CONST.   OBLIQUITY. 

D. 

F. 

D. 

F. 

MAX.   OBLIQUITY. 

0°  0'   0 

5.58 

CO 

5.58 

00 

0°  0    0 

2°  28'  42" 

6 

146 

6 

144 

2°  30' 

7°    0'  47" 

7 

52 

7 

50 

7°  12' 

10°  17'    4" 

8 

35 

8 

34 

10°  35'  18" 

13°  45'    5" 

9 

28 

9 

27 

13°  20' 

14°  40'  33" 

10 

24 

10 

23 

15°  89'    8" 

15°  54'  56" 

11 

22 

11 

21 

17°    8'  34" 

17°  26'  26" 

12 

20 

12 

19 

18°  56'  50" 

18°  17'  57" 

13 

19 

13 

18 

20° 

19°    2'  37" 

14 

18 

14 

17 

21°  10'  35" 

20°  17'    4" 

15 

17 

15 

16 

22°  80' 

21°  26'  24" 

18 

16 

17 

15 

24° 

22°  43'  40" 

20 

15 

20 

14 

25°  42'  51" 

24°  10'  12" 

25 

14 

24 

13 

27°  41'  32" 

25°  47'  45" 

34 

13 

32 

12 

30° 

27°  38'  12" 

59 

12 

57 

11 

32°  43'  38" 

29°  44'    6" 

515 

11 

382 

10 

36° 

30° 

00 

10.88 

00 

9.83 

36°  37'  21' 

INVOLUTE   AKD   EPICYCLOID   COMPAEED. 

Limiting  Numbers  of  Teeth. 


163 


EECESS  =  J  PITCH.                                    TOOTH  =  SPACE. 

INVOLUTE. 

EPICYCLOID  AL. 

CONST.    OBLIQUITY. 

D. 

F. 

D. 

F. 

MAX.   OBLIQUITY. 

3.25 

CO 

o°  o'  <r 

16°    240" 

546 

CO 

4 

34 

7°.  56'  28" 

17°  59'     6" 

6 

15 

5 

19 

14°  54'    4" 

20°  43'  34" 

7 

13 

6 

14 

19°  17'    8" 

22°  43'  27" 

8 

12 

7 

12 

22°  30' 

24°  14'  35" 

9 

11 

8 

11 

24°  32'  44" 

25°  13'  56" 

10 

10 

9 

10 

27° 

27°  38'  12" 

13 

9 

11 

9 

30° 

30°  29'  58" 

20 

8 

16 

8 

33°  45' 

33°  56'  54" 

70 

7 

33 

7 

38°  34'  16" 

35°  46   57  " 

CO 

6.54 

oo 

6.35 

42°  31'  11" 

BECESS  —  f  PITCH.                                   TOOTH  =  SPACE. 

INVOLUTE. 

EPICYCLOIDAL. 

CONST.   OBLIQUITY. 

k 

D. 

F. 

D. 

F. 

MAX.   OBLIQUITY. 

2,58 

00 

0°    0'    0" 

3 

37 

6°  29'  11" 

4 

15 

16° 

21°  32'  59  " 

5.30 

CO 

5 

11 

21°  49'    6" 

23°  38'  20" 

6 

10 

6 

10 

24° 

24°  57'  30" 

7 

9 

7 

9 

26°  40' 

27°  38'  12" 

9 

8 

8 

8 

30° 

30°  53'  47" 

12 

7 

11 

7 

34°  17'    9" 

34°  55'  12" 

28 

6 

21 

6 

.40° 

37°  58  55" 

oo 

5,36 

00 

5.17 

46°  25'  18" 

164 


INVOLUTE   AtfD   EPICYCLOID    COMPARED. 


277.  But   the   labor   of   computing  these   tables   was   undertaken 
mainly  with  the  object  of  making  a  fuller  comparison  than  has  before 
been  made  between  the  two  leading  systems  of  spur  gearing  :  and  to 
facilitate  this,  the  limiting  numbers  and  th3  maximum  obliquities  for 
epicycloidal,  are  here  given  side  by  side  with  those  for  involute  teeth. 

One  striking  point  of  difference  will  at  once  be  noticed.  In  the 
epicycloidal  system,  as  the  arc  of  recess  diminishes,  the  least  number 
for  the  driver  also  diminishes,  and  the  obliquity  is  always  zero  at  the 
limit  where  the  follower  becomes  a  rack. 

In  the  involute  system,  on  the  other  hand,  this  obliquity  rapidly 
increases,  while  the  number  for  tho  least  driver  remains  practically 
the  same,  varying  but  little  even  at  the  limit. 

The  reason  is  not  far  to  seek.  The  generating  circle  for  the  driving 
face  of  the  epicycloidal  tooth,  which  alone  has  to  do  with  the  reced- 
ing action,  is  different  from  and  wholly  independent  of  that  for  the 
flank,  which  produces  the  required  amount  of  approaching  action. 

But  the  outline  of  the  involute  tooth  is  one  continuous  curve,  and 
the  rectilinear  generatrix,  or  line  of  action,  is  the  same  for  the  part 
without  the  pitch  circle  and  the  part  within.  The  sum  of  these  two 
segments  must  always  be  at  least  equal  to  the  length  of  the  pitch  arc 
on  the  base  circle.  If  then  the  outer  one,  upon  which  the  arc  of  re- 
cess depends,  be  decreased,  the  inner  one  must  be  correspondingly 
increased  ;  and  this  obviously  involves  a  greater  obliquity. 

278.  The  difference  between  the  two  systems  is  best  illustrated  by 
constructing,  as  in  Fig.  1GO,  the  nomodont  for  each  under  like  condi- 
tions.    The  arc  of 
recess  is  here  taken 
at  §  the  pitch,  and 
the  scale  for  the  ab- 
scissas is  twice  that 
for   the   ordinates : 
the  distances  of  the 
vertical    and  hori- 
zontal    asymptotes 
from  the  axes  being 
respectively       5.30 
and   5.36   for  AA, 

which  represents  the  involute  system,  and  2.58  and  5.17  for  BB,  the 
curve  for  the  epicycloidal  system. 

The  law  of  variation  in  the  obliquities  is  also  most  clearly  exhibited 
by  the  same  graphic  means  :  thus,  the  ordinates  of  the  curves  CC  and 
DD  arc  proportional  to  the  tabular  values  of  the  obliquity  for  invo- 


SECOND    GENERATION   OF   INVOLUTE. 


165 


lute  and  epicycloidal  teeth  respectively,  the  abscissas,  as  before,  rep- 
resenting the  numbers  of  teeth  of  given  drivers. 

These  ordinates  are  drawn  to  the  same  scale  as  those  of  the  nomo- 
donts,  the  numerical  value  being  in  each  case  the  one-hundredth  part 
of  the  obliquity  taken  to  the  nearest  minute.  Thus,  the  first  ordinate 
for  C  is  the  vertical  asymptote  to  A,  and  its  value  is 


As  the  number  for  the  driver  approaches  infinity,  the  obliquity  ap- 
proaches the  limit  37°  59',  as  seen  by  reference  to  the  table,  and 

37°  59'  -  22'79 

~ioo"     23'79' 

at  which  distance  above  the  axis  of  abscissas  this  curve  will  have  a 
horizontal  asymptote. 

In  like  manner,  we  find  that  the  curve  DD  has  also  a  horizontal 
asymptote,  at  a  distance  of  27.85  above  the  axis  of  abscissas.  But 
this  curve,  instead  of  beginning  abruptly  like  the  other,  and  inter- 
secting the  vertical  asymptote  to  BE,  is  tangent  to  that  line  at  its  foot. 

279.  The  Involute  Generated  by  Rolling  Contact  with  the  Pitch  Cir- 
cle. —  Before  dismissing  the  invo- 
lute, it  remains  to  prove  that  it 
can  be  generated  in  such  a  man- 
ner as  to  accord  with  the  general 
principle  stated  in  (181). 

In  Fig.  161,  let  A  The  the  com- 
mon tangent  of  the  pitch  circles, 
and  let  CQ,  DP,  the  radii  of  the 
base  circles,  be  perpendicular  to 
the  line  of  action  PQ,  drawn  at 
pleasure.  Produce  DP  to  inter- 
sect A  T  in  E;  and  let  POA  be  a 
logarithmic  spiral,  of  which  P  is 
the  pole  and  A  T  the  tangent  at 
the  extremity  of  the  radiant  PA. 
If  now  the  portion  AP  of  this 
spiral  roll  upon  AE,  the  pole  P 
will  trace  the  right  line  PE, 
which,  therefore,  may  represent 
the  direction  and  velocity  of  the  FlG-  161« 

motion  of  P  due  to  this  rolling.     With  this  motion  let  there  be  com- 


166  SECOND   GENEBATION   OF   INVOLUTE. 

pounded  a  motion  similarly  represented  by  PF,  equal  and  parallel  to 
EA,  then  the  resultant  is  PA  ;  the  effect  being  the  same  as  if,  during 
the  rolling  of  the  spiral  upon  the  tangent,  the  tangent  should  move 
to  the  right  as  shown  by  the  horizontal  arrow.  Suppose  further  that 
the  pitch  circles  meantime  move  in  rolling  contact  with  the  tangent ; 
they  will  then  have  turned  about  the  fixed  centres  C  and  J9,  through 
angles  measured  by  the  arcs  AM,  AL,  equal  to  each  other  and  to  AE9 
and  the  rectilinear  motions  PE,  PF,  being  uniform,  these  rotations 
will  also  be  uniform.  Also,  the  pitch  circles  will,  during  this  rota- 
tion, move  in  rolling  contact  with  the  spiral  AP,  which  being  taken 
as  the  describing  line,  will  carry  the  pole  P  in  the  right  line  PA  with 
uniform  velocity.  But  we  have  already  seen  that  such  motion  of  P 
will  simultaneously  trace  upon  the  planes  of  the  two  wheels,  the  in- 
volutes AH  and  AI  of  the  base  circles.  And  in  like  manner  it  may 
be  shown  that  the  continuations  AJ  and  AK  of  these  involutes  may 
be  generated  by  the  similar  and  equal  spiral  ANQ,  rolling  in  contact 
with  the  pitch  circles,  its  pole  Q  being  the  tracing  point. 


CHAPTER  X. 


SPUE   GEAEIKG — CONTINUED. 


Conjugate  Teeth.  Sang's  Theory.  Path  of  Contact.  Unsymmetrical  Teeth 
Approximate  Forms.  The  Odontographs.  Diametral  Pitch.  Manufacture 
of  Gear  Cutters.  Determination  of  Series  of  Equidistant  Cutters. 

Conjugate  Teeth. 

280.  It  follows  from  the  general  principles  stated  in  (181),  that  if 
a  tooth-outline  of  any  reasonable  form  be  assumed,  it  may  be  gener- 
ated by  some  describing  curve  rolling  upon  the  pitch  circle ;  and  if 
that  describing  curve  be  determined,  and  used  in  connection  with  any 
other  pitch  circle,  the  tooth  thus  formed  will  evidently  gear  correctly 
with  the  assumed  one.  Teeth  thus  related  are  said  to  be  conjugate  to 
each  other ;  since  had  the  second  one  been  given,  the  application  of 
the  above  process  would  have  resulted  in  the  formation  of  a  tooth 
identical  with  the  first.  • 

Now  it  may  be  required  to  lay  out  a  wheel  to  gear  with  one  already 
made ;  and  supposing  the  original  drawing  to  be  lost,  the  tooth  of 
this  given  wheel  is  to  all  intents  and  purposes  an  assumed  one :  a 
ready  means  of  tracing  its  conjugate  is  therefore  of  some  practical 
importance. 

The  radius  of  the  original  pitch  circle,  even,  may  be  unknown,  but 
this  does  not  affect  the  above  reasoning.  We  must  assume  a  radius  : 
and  if  it  should  not  be  the  same  as  the  one  originally  used,  the  form 
of  the  describing  curve  will  be  modified  accordingly,  and  the  outline 
of  the  conjugate  tooth  will  still  be  correct. 

The  form  of  the  describing  curve,  which,  by  rolling  on  the  assumed 
pitch  circle  will  generate  the  given  tooth,  can  in  general  be  only  ap- 
proximately determined  by  graphic  processes  (See  Appendix).  And 
even  if  it  were  susceptible  of  exact  construction,  the  method  above 
indicated,  of  ascertaining  by  means  of  it  the  outline  of  the  conjugate 
tooth,  involves  much  greater  labor,  and  is  withal  less  satisfactory  in 


168 


CONJUGATE  TEETH. 


FIG.  162. 


all  practical  respects,  than  the  simple  and  direct  mechanical  expedient 
illustrated  in  Fig.  162. 

281.  Let  the  form  of  a  tooth,  T,  of  the  given  wheel,  be  accurately 
cut  out  of  a  piece  of  sheet  metal,  KG,  which  is  properly  centred  to 

turn  upon  a  fixed  pin  at  D ;  and  let 
AD  be  the  radius  of  the  assumed  pitch 
circle. 

Behind  this  is  another  piece  of  sheet 
metal  MN9  being  a  portion  of  a  disk 
turning  upon  a  fixed  centre  C.  Since 
the  radius  AD  and  the  number  of  teeth 
upon  the  given  wheel  are  known,  the 
radius  A  (7  is  determined  by  the  velocity 
ratio  required. 

The  outline  of  the  tooth,  T9  is  now  to 
be  traced  on  the  blank  disk,  MN9  with 
a  fine  marking  point :  after  which  let 
each  piece  be  rotated  through  a  small 
angle,  as  though  the  pitch  circles  moved  in  rolling  contact.  The 
proper  relative  amounts  of  these  angular  motions  are  easily  regulated 
by  means  of  subdivisions,  on  the  circular  edges,  in  connection  with 
fixed  marks  on  the  board  to  which  this  mechanism  is  secured,  as 
shown  at  0  and  L.  The  outline  of  T  is  then  to  be  traced  again  ;  and 
by  repeating  this  process  the  form  of  the  conjugate  tooth  may  be  very 
accurately  mapped  out,  since  each  of  the  marks  thus  made  upon  the 
blank  MN9  by  the  very  nature  of  the  operation,  is  tangent  to  its  out- 
line. 

Moreover,  if  both  sides  of  T  be  traced  upon  the  disk  MN  in  its 
various  positions,  thus  determining  the  boundaries  of  the  space  be- 
tween two  adjacent  conjugate  teeth  S9  S'9  there  will  be  no  backlash  ; 
for  by  construction  the  backs  as  well  as  fronts  of  the  teeth  are  in  con- 
tact at  all  times.  It  is  true  that  the  tooth  and  the  space  of  the  given 
wheel  may  not  be  measured  by  equal  arcs  upon  the  pitch  circle,  which 
was  drawn  at  pleasure.  Nor  is  this  essential,  since  the  spaces  and  the 
teeth  of  the  derived  wheel  will  be  correspondingly  unequal ;  the  total 
pitch  arc,  or  sum  of  a  tooth  and  a  space,  being  however  necessarily 
the  same  for  both  wheels. 

Sang9 s  Theory  of  the  Teeth  of  WJieels. 

282.  By  giving  different  values  to  AC  in  Fig.  162,  we  may  construct 
a  whole  series  of  wheels,  having  various  numbers  of  teeth,  all  working 
properly  with  the  given  wheel. 


169 

Next,  let  the  tooth  of  any  one  of  these  be  taken  as  the  assumed 
tooth  in  the  above  process,  and  used  in  like  manner  with  various  pitch 
circles,  thus  forming  a  second  series  of  wheels. 

From  the  mode  of  generation  it  is  evident  that  any  wheel  belonging 
to  either  series  will  gear  correctly  with  any  one  of  those  constituting 
the  other  series.  But  the  wheel  of  a  given  number  of  teeth  in  the 
first  may  not  be  like  that  one  of  the  second  which  has  the  same  num- 
ber. If  the  outlines  of  conjugate  teeth  upon  any  two  wheels  of  the 
same  number  be  made  alike  by  any  means,  it  is  clear  that  the  two 
series  will  be  identical,  and  all  the  wheels  of  both  will  be  inter- 
changeable. 

Now  in  the  derivation  of  either  series,  the  assumed*  tooth  always 
occupies  and  maps  out  the  space  between  two  adjacent  conjugate 
teeth ;  and  this  holds  true  when  the  latter  becomes  infinite.  From 
this  it  follows  that  the  racks  of  the  two  series  are  in  any  case  exactly 
converse  ;  their  contours  are  identical,  the 
teeth  of  the  one  being  precisely  like  the 
spaces  of  the  other,  and  vice  versa,  as  shown 
in  Fig.  163. 

When,  therefore,  the  two  series  become 
identical,  the  two  racks  will  be  identical ; 
each  will  be  its  own  converse,  and  have  its  FlG-  163- 

teeth  and  its  spaces  similar  and  equal  to  each  other.  This  requires 
that  the  tooth  and  space  shall  be  equal  as  measured  on  the  pitch  line, 
and  also  that  the  portions  «,  Z>,  of  the  contour,  shall  be  respectively 
similar  and  symmetrical  to  the  portions  a'  l>  on  the  opposite  side  of 
the  pitch  line. 

And  if,  as  all  along  supposed,  the  teeth  are  symmetrical  to  a  radius, 
that  is,  have  their  fronts  and  backs  alike,  it  is  further  necessary  that 
a  and  b  shall  be  similar  and  equal. 

283.  Now  it  is  evident  that  in  the  process  of  Fig.  162,  we  may 
assume  the  tooth  of  a  rack,  and  from  it  derive  the  conjugate  teeth  of 
a  series  of  wheels,  by  a  modification  in  detail  which  is  too  obvious  to 
require  explanation. 

And  from  the  foregoing,  the  deduction  is  that  if  this  original  rack- 
tooth  be  bounded  by  four  similar  and  equal  lines  in  alternate  rever- 
sion, the  wheels  whose  teeth  are  thus  determined  will  form  an  inter- 
changeable set.  It  will  readily  be  seen  that  if  the  rack  be  composed 
of  cycloidal  arcs,  the  result  will  be  the  familiar  epicycloidal  system 
with  a  constant  describing  circle  ;  while  if  it  be  bounded  by  oblique 
right  lines,  the  involute  system  is  at  once  reproduced. 

The  contour  of  the  rack  being  arbitrary,  with  the  limitation  just 


170 


TO   FIND  THE   PATH  OF   CONTACT. 


mentioned,  any  number  of  systems  may  be  thus  derived,  each  differ- 
ing from  the  others  in  the  peculiarities  of  its  action,  more  particularly 
in  respect  to  the  variation  in  the  obliquity. 

Professor  Edward  Sang,  in  a  most  elaborate  treatise,  has  discussed 
this  method  of  constructing  the  teeth  of  wheels,  which  consists  sub- 
stantially in  reversing  the  usual  order  of  proceeding.  His  treatment 
of  the  subject  is  analytical  and  abstruse,  the  style  obscure  and  pedan- 
tic to  the  last  degree ;  and  although  the  theory  of  gearing  is  fully 
developed,  the  equally  important  practical  operations  of  laying  it  out 
are  very  much  neglected. 

284.  Determination  of  the  Path  of  Contact. — The  theory  is,  however, 
more  comprehensively  seen  from  this  point  of  view  than  from  any 
other,  since  the  important  part  played  by  the  path  of  contact  is 
brought  more  prominently  to  our  notice  :  and  Professor  Sang's  inves- 
tigations relate  more  particularly  to  the  form  of  that  path,  as  depend- 
ent upon  the  contour  of  the  assumed  tooth. 

It  is  not  necessary  to  know  the  describing  curve  in  order  to  deter- 
mine the  path  of  contact.  In  Fig.  164,  let  A  B  be  the  face  of  a  tooth 

for  the  wheel  whose  centre  is  D ;  draw 
a  series  of  normals  to  this  face,  cutting 
the  pitch  circle  at  1,  2,  etc.  During 
the  action,  the  common  normal  of  the 
tooth-outlines  must  at  every  instant  pass 
through  the  common  point  of  the  pitch 
circles  ;  therefore  as  the  points  1,  2,  etc., 
successively  reach  the  line  of  centres,  the 
normals  1«,  2b,  oc,  will  take  the  posi- 
tions Aa'9  Ah ',  Ac ',  thus  determining 
the  path  of  contact  AB '. 

Let  C  be  the  centre  of  the  engaging 
wheel ;  then  the  form  of  the  conjugate 
flank  may  also  be  found  without  making 
use  of  the  describing  curve.  For  con- 
sidering the  given  wheel  as  the  driver, 
the  angular  motions  of  the  follower  cor- 
responding to  the  already  ascertained  an- 
gular motions  aaf,  ~bl> ,  etc.,  of  the  driver, 
may  be  determined,  since  the  velocity 
ratio  is  known.  Arcs  a'a",  W',  c'c", 
FIG.  164.  measuring  these  angular  motions  of  the 

follower,  then,  are  to  be  set  off  on  circles  of  which  C  is  the  centre,  and 
the  curve  AE  thus  determined  is  the  acting  conjugate  flank ;  which 


PATH   OF   CONTACT.  171 

will,  of  course,  require  to  be  extended  as  shown  in  dotted  lines  for  the 
formation  of  a  clearing  space,  as  usual. 

285.  But  although  it  is  thus  possible  to  determine  the  path  of  con- 
tact and  the  conjugate  tooth  by  graphic  means,  this  method  is  not  in 
general  well  suited  for  ordinary  practical  operations,  since  it  involves 
the  drawing  of  the  normals  to  a  curve  whose  nature  may  not  be  known, 
and  this  of  itself  (see  Appendix,  (7)  is  a  matter  of  considerable  labor  ; 
while  the  whole  process  above  described  is  not  only  tedious,  but  re- 
quires most  careful  manipulation  in  order  to  secure  satisfactory  results. 
Consequently,  if  the  form  of  a  tooth  be  given  or  assumed,  the  me- 
chanical method  of  Fig.  162  is  for  practical  purposes  by  far  the  best 
for  determining  that  of  its  conjugate.    It  is  true  that  it  gives  no  indi- 
cation of  the  path  of  contact,  and  therefore  does  not  enable  us  to 
ascertain  precisely  either  where  the  action  begins  or  where  it  ends. 
But  if  neither  wheel  has  less  than  twelve  teeth,  and  these  of  propor- 
tions in  accordance  with  any  of  the  arbitrary  rules  given  in  (201), 
there  can  be  no  question  that  the  total  angle  of  action  will  be  ample. 

For  when  we  consider  that  in  the  involute  system  the  path  of  con- 
tact is  a  right  line  joining  the  initial  and  terminal  points  of  action, 
while  in  the  epicycloidal  it  is  composed  of  two  circular  arcs  tangent 
to  each  other  and  to  both  pitch  circles  at  their  common  point,  and 
also  that  it  must  pass  through  this  last-mentioned  point  in  all  cases, 
it  will  appear  more  than  likely  that  with  any  other  reasonable  form 
of  tooth,  this  path  will  be  intermediate  between  these  extremes.  And 
if  so,  the  difference,  either  in  respect  to  the  limiting  numbers  of  teeth 
or  the  variations  in  the  obliquity  of  action,  is  greater  between  these 
than  between  any  other  two  systems  of  gearing  :  and  either  of  them, 
as  before  stated,  will  with  the  above-mentioned  numbers  and  propor- 
tions give  results  practically  satisfactory  for  most  uses. 

It  is  beyond  the  scope  of  this  work  to  discuss  in  detail  the  various 
forms  of  teeth  which  may  be  constructed  by  the  application  of  Sang's 
Theory.  In  all  the  processes  of  "  laying  out "  gearing  upon  the  draw- 
ing-board, the  simplicity  of  the  paths  of  contact  and  of  the  describing 
lines,  peculiar  to  the  involute  and  the  epicycloidal  systems,  gives 
them  a  great  and  obvious  advantage  over  others  ;  while  years  of  expe- 
rience have  established  the  fact  that  in  practical  operation  these  forms 
have  no  superiors. 

286.  Unsymmetrical  Teeth. — Thus  far,  the  teeth  have  been  supposed 
to  be  symmetrical,  that  is,  to  have  their  fronts  and  backs  alike,  as 
indeed  they  are  made  except  in  rare  instances.     But  this  is  not  at  all 
a  matter  of  necessity,  and  under  some  circumstances  it  may  be  advan- 
tageous to  make  them  otherwise.     For  example,  in  Fig.  165,  the 


172 


UNSYMMETRICAL  TEETH. 


acting  outlines  (the  lower  wheel  being  supposed  to  drive  to  the  right) 

are  of  the  epicycloidal  form ;  the 
describing  circles  are  large,  in  order  to 
reduce  the  obliquity  to  a  minimum, 
and  were  the  tooth  made  of  the  same 
form  on  the  back,  this  might  render 
it  too  weak  at  the  root.  But  if  the 
motion  is  never  to  be  reversed,  this 
weakness  may  be  avoided  by  making 
the  back,  as  shown,  an  involute  of 
considerable  obliquity. 

The  conjugate  racks,  derived  by  the 


FIG.  165.  FIG.  166. 

process  of  Fig.  162  from  a  tooth  of  this  description,  would  be  of  the 
forms  shown  in  Fig.  166;  which,  it  will  be  observed,  are  identical, 
notwithstanding  their  lack  of  symmetry. 


Approximate  Forms  of  Teeth. 

287.  For  the  perfect  working  of  mechanism  in  which  wheels  are 
employed,  it  is  unquestionable  that  the  forms  of  the  teeth  should  be 
exactly  correct.  And  if  they  are  to  be  cut,  there  is  no  reason  why 

the  teeth  should  not  be  laid  out  with  the 
utmost  precision,  since,  the  cutter  once 
made,  it  is  as  easy  to  cut  one  shape  as 
another. 

But  for  ruder  machinery,  when  the 
wheels  are  simply  to  be  cast,  so  great  ac- 
curacy may  be  sometimes  thought  unnec- 
essary ;  and,  especially  if  time  presses,  a 
more  ready  means  of  securing  a  reasonably 
close  approximation  may  be  desirable. 

In  Fig.  167,  draw  through  /,  the  com- 
mon point  of  the  pitch  circles  XY,  VW, 
FIG.  167.  a  line  of  action  A  IP  at  pleasure,  and  IE 

perpendicular  to  it.     Draw  CA  at  pleasure,  cutting  IE  in  E,  and 
join  ED  by  a  right  line  cutting  AP  in  B. 


APPROXIMATE   TOOTH-FORMS.  173 

Now  if  CA,  DB,  be  a  pair  of  levers  joined  by  a  link  AB,  then, 
letting 

v   —  ang.  vel.  about  D, 
v   =     "      "        "      C, 
we  have  the  value 

v_  _CIf 
v'  ~DV 

and  since  by  the  construction  E  is  the  instantaneous  axis,  this  velocity 
ratio  will  be  momentarily  constant.     (See  Figs.  39  and  40.) 

If  now  through  any  point  P  on  the  line  of  action  we  describe  two 
circular  arcs,  one  about  B,  as  a  tooth-face  for  the  lower  wheel,  the 
other  one  with  centre  A  being  the  conjugate  flank  ;  the  velocity  ratio 
determined  by  them  will  at  the  instant  be  exactly  the  same  as  that  of 
the  combination  of  link  and  levers,  and  will  not  differ  appreciably 
from  it  during  an  elementary  motion  in  either  direction. 

288.  The  above  argument,  it  will  be  observed,  does  not  depend 
either  upon  the  obliquity,  or  upon  the  position  of  the  point  P,  both 
of  which  are  optional. 

But  in  order  to  reduce  the  application  of  this  general  principle  to 
a  system,  let  Iff,  IK,  be  each  one  half  the  pitch  arc,  and  let  IP  be 
an  arc  of  the  same  length  upon  a  describing  circle  ;  then  P/will  be 
the  common  normal  to  an  epicycloid  KP  and  a  hypocycloid  HP,  tan- 
gent at  P}  and  these  are  correct  tooth-outlines.  Let  B  and  A  be  the 
centres  of  curvature  for  these  two  curves  respectively,  at  the  common 
point  P  ;  the  circular  arcs  may  then  be  used  as  approximating  closely 
enough  to  the  exact  forms  for  many  practical  purposes,  provided  that 
the  obliquity  be  not  too  great  nor  the  teeth  too  long. 

And  if  the  same  describing  circle,  and  the  same  arc  IP,  be  used, 
the  obliquity  of  the  line  of  action,  for  the  mean  position  when  the 
curves  are  in  contact  at  P  and  the  velocity  ratio  is  exactly  correct, 
will  be  the  same  for  all  pitch  circles. 

289.  Willis's  Odontograph. — The  system  above  explained  was  origi- 
nated by  Professor  Willis,  and  upon  it  is  based  the  invention  of  his 
well  known  Odontograph.     Making  the  arc  IP  —  30°,  the  obliquity 
is  15°,  and  the  angle  DIA  =  75°.     Then  having  calculated  the  dis- 
tances from  /  of  the  centres  of  curvature  at  P  for  the  tooth-faces  and 
flanks  for  various  pitches  and  numbers  of  teet^h,  these  are  tabulated, 
and  the  instrument  is  constructed  as  in  Fig.  168,  which  also  illus- 
trates the  method  of  using  it. 

It  consists  merely  of  two  arms  of  German  silver,  making  an  angle 
of  75°  with  each  other,  the  edge  of  one  corresponding  to  the  radius 


174 


WILLIS'S  ODOtfTOGRAPH. 


Fio.  168. 


LI,  that  of  the  other  to  the  line  of  action  AP  in  Fig.  167  ;  and  this 
latter  edge  is  divided  into  a  scale  of  equal  parts,  numbered  both  ways 
from  the  intersection  of  the  two  edges,  which  corresponds  to  the 
point  /.  We  have  thus  two  scales,  the  one  on  the  right  being  the 
"  Scale  of  Centres  for  Faces,"  that  on*  the  left  for  flanks  ;  and  the 
table  above  mentioned  has  two  corresponding  divisions. 

Suppose  it  to  be  required  to  lay  out 
a  wheel  of  20  teeth,  the  circular  pitch 
being  1J  inches.  These  data  deter- 
mine the  radius  of  the  pitch  circle  VW, 
Fig.  168;  upon  which  set  off  the  pitch 
arc  EF,  bisect  it  at  L,  and  draw  the 
radii  DE  and  DF.  Set  the  odonto- 
graph so  that  the  slant  edge  coincides 
with  DE,  the  point  E  thus  coming 
just  at  the  zero  of  the  scales.  Then  in 
the  "Table  of  Centres  for  Flanks,"  in 
the  column  for  1^-inch  pitch,  opposite 
20  in  the  column  of  "Numbers  of  Teeth,"  we  find  the  number  37. 

The  position  indicated  by  this  number  on  the  "  Scale  of  Centres 
for  Flanks,"  fixes  the  point  A  as  the  centre  of  the  circular  arc  through 
L,  which  gives  the  approximate  form  of  the  required  flank.  Then 
setting  the  odontograph  in  like  manner  by  the  radius  DF,  by  a  similar 
process  we  find  on  the  other  scale  the  location  of  the  centre  B  of  the 
circular  arc  through  L,  which  forms  the  face  of  the  tooth. 

290,  It  is  evident  that  wheels  of  the  same  pitch,  laid  out  by  this 
odontograph,  are  interchangeable,  their  teeth  being  in  effect  epicy- 
cloidal  with  a  constant  describing  circle.  Since  half  the  pitch  arc 
measures  30°  upon  this  circle,  the  wheel  of  12  teeth,  the  least  for 
which  the  distinguished  inventor  designed  it  to  be  used,  will  have 
radial  flanks. 

Of  course  the  approximating  circular  arcs  will  deviate  more  and 
more  from  the  true  curves  as  the  tooth  is  made  longer,  and  will  not 
answer  at  all  for  low-numbered  pinions.  But  if  the  length  of  the 
tooth  be  limited  by  the  arbitrary  rules  given  in  (201),  the  instrument 
may  be  used  with  confidence  for  all  ordinary  purposes  when  the 
wheels  are  to  be  cast,  and  none  have  less  than  12  teeth.  In  the  case 
of  a  rack,  the  radii  DE,  t)F,  of  Fig.  168  will  become  perpendicular 
to  the  pitch  line  ;  in  laying  out  annular  wheels  the  positions  of  face 
and  flank  are  transposed,  and  due  attention  must  be  paid  to  the  limits 
in  regard  to  the  diameters  of  the  pitch  circles,  which  have  already 
been  discussed. 


TEMPLET   ODOKTOGKAPH. 


175 


It  is  also  to  be  remarked  that  the  tables  which  accompany  the  in- 
strument do  not  contain  radii  of  curvature  for  all  the  consecutive 
numbers  of  teeth.  The  variation  in  the  correct  contour,  due  to  the 
addition  of  a  single  tooth,  rapidly  diminishes  as  the  actual  number 
increases,  so  that  practically  the  same  outline  serves  for  the  teeth  of 
several  different  wheels.  If  then  the  number  assigned  be  not  found 
in  the  table,  the  nearest  number  to  it  is  to  be  taken  instead. 

Also,  if  the  given  pitch  be  one  not  inserted  in  the  tables,  the  radii 
required  may  be  found  by  direct  proportion  from  those  of  other 
pitches  which  are  tabulated  ; — for  4-inch  pitch  by  doubling  the  radii 
for  2-inch  pitch,  halving  those  given  for  1J-  inch  pitch  in  order  to 
find  the  radii  for  |-inch  pitch,  and  so  on,  as  occasion  may  require. 


Robinson's  OdontograpJi. 

291.  This  differs  entirely  from  the  preceding,  both  in  principle 
and  in  the  manner  of  using  it.  In  Fig.  169,  let  VW  be  an  arc  of 
the  pitch  circle,  to 
which  CTis  tangent  at 
A,  the  middle  point  of 
the  tooth,  A  0,  being  the 
half  thickness.  Then 
the  odontograph  is  to  be 
so  set  that  CT  shall  not 
only  be  tangent  to  tho 
lower  curved  edge  BFH, 
but  also  cut  the  grad- 
uated edge  BE,  which 
must  pass  through  0,  at 
a  certain  division  C,  determined  by  the  aid  of  tables  and  instructions 
which  accompany  the  instrument.  The  edge  CE  is  then  used  as  a 
curved  ruler  for  drawing  the  face  of  the  teeth,  after  which  the  odonto- 
graph, being  made  of  thin  metal  and  graduated  on  both  sides,  is 
turned  over,  and  the  opposite  face  of  the  tooth  is  drawn  in  like  man- 
ner. A  different  (t setting  number"  is  then  determined  from  the 
tables,  which  brings  a  different  portion  of  the  curved  edge  into  posi- 
tion for  describing  the  flanks  of  the  teeth.  By  certain  modifications 
in  the  determination  of  the  setting  numbers  and  in  the  placing  of  the 
instrument,  involute  teeth  also  can  be  drawn,  as  well  as  epicycloidal 
ones  with  various  describing  circles,  in  either  outside  or  inside  gear. 

This  remarkably  ingenious  invention,  which  has  a  much  wider  and 
more  flexible  range  of  action  than  that  of  Willis,  is  called  the  Tern- 


FIG. 


176  DIAMETKAL   PITCH. 

plet  Odontograph,  from  the  fact  that  the  forms  of  the  teeth  are  drawn 
by  it  directly  ;  and  if  it  be  desired  to  lay  out  the  whole  wheel,  its  use 
is  much  facilitated  by  attaching  it  temporarily  to  a  wooden  rod  or 
radius  bar,  turning  about  the  centre  of  the  wheel. 

292.  Its  construction,  obviously,  depends  upon  the  finding  of  a 
curve  for  the  graduated  edge,  of  rapidly  changing  curvature,  and  of 
such  a  nature  that  its  different  portions  shall  closely  approximate  to 
the  initial  parts,  or  those  next  the  pitch  circles,  of  teeth  of  all  pitches 
and  sizes,  of  either  the  involute  or  epicycloidal  forms.     This  very 
peculiar  property  Prof.  Bobinson  has  shown,  by  a  long  and  elaborate 
investigation,*  to  be  possessed  by  a  certain   logarithmic  spiral,  of 
which  form,  accordingly,  the  working  edge  BCE  is  made.     The  other 
curve  BFH,  it  may  be  added,  is  the  evolute  of  BCE,  and,  therefore, 
a  similar  and  equal  spiral. 

Diametral  Pitch. 

293,  In  designing  and  laying  out  wheels  upon  any  of  the  systems 
described,  it  is  necessary  to  find  the  circular  pitch,  to  which  only 
reference  has  thus  far  been  made  ;  not  only  because  it  is  used  in  the 
graphic  constructions,  but  because  the  strength  of  the  tooth  depends 
upon  its  thickness. 

Were  there  nothing  to  the  contrary,  it  would  be  most  convenient 
to  express  the  pitch  in  whole  numbers  or  in  manageable  fractions,  as 
2-inch  pitch,  J-inch  pitch,  and  so  on.  But  since  for  complete  wheels 
the  circular  pitch  must  be  an  aliquot  part  of  the  circumference,  the 
diameters  of  the  pitch  circles  will  often  contain  awkward  decimals  if 
this  plan  be  adhered  to  ;  and  practically  it  is  much  more  important 
to  have  the  diameter  a  whole  number,  or  a  convenient  fraction,  than 
that  the  circular  pitch  should  be  either  the  one  or  the  other. 

This  consideration  has  led  to  the  introduction,  and  the  almost  ex- 
clusive adoption  for  cut  gearing,  of  what  is  called  the  Diametral  Pitch ; 
which  is  merely  the  quotient  found  by  dividing  the  diameter  of  the 
pitch  circle,  instead  of  its  circumference",  into  as  many  equal  parts  as 
there  are  to  be  teeth  upon  the  wheel  under  consideration  : 

whence, 

—  i  T»-J.  i  Diameter  Circular  Pitch 

Diametral  Pitch  =  =r= ^-=~ —  r  >  =  -  -  > 

No.  of  Teeth  3.1416 

and 

Circular  Pitch  =  Diametral  Pitch  x  3.1416. 

*  See  Van  Nostrand's  Eclectic  Engineering  Magazine,  July,  1876. 


DIAMETRAL   PITCH. 

294.  In  the  use  of  this  system,  convenient  values  ol 
pitch  are  selected,  each  being  a  fraction  with  unity  for  its  numerator 
and  an  integer  for  its  denominator,  as  1,  J,  ],  -J-,  ^  3^,  etc. 

The  denominators  of  these  fractions  only  are  commonly  used  in 
giving  the  diametral  pitch;  thus  an  "  8-pitch  wheel "  is  one  which 
has  eight  teeth  for  each  inch  of  diameter,  or  whose  diametral  pitch 
is  J".  This  is,  in  fact,  merely  inverting  the  fraction,  and  giving  the 

» No.  of  Teeth 

value  of  • — ^ —     ;  thus,  let  a  wheel  of  16  inches  diameter  have 

Diameter 

80  teeth  ;  then,  -J$  =  -J"  —  diametral  pitch,  but  |$  =  5,  and  we  call 
it  a  "  5-pitch  "  wheel.     By  this  system  the  calculations  as  to  diameter 
and  number  of  teeth  are  made  very  simple  ;  as,  for  example  : 
Required,  the  diameter  of  a  4-pitch  wheel  with  37  teeth  : 

^--c=  9J  =  diameter. 

How  many  teeth  of  16-pifcch  on  a  wheel  of  3J  diameter  ? 
3|  x  16  =  G2  =  No.  teeth. 

The  tooth  may  be  made  to  project  outside  the  pitch  circle  a  definite 
fraction  of  the  diametral  pitch,  as  in  the  case  of  the  circular  pitch  ; 
and  thus  the  size  of  the  blank  may  be  readily  ascertained.  If  this 
projection  be  made,  for  instance,  1^  the  diametral  pitch,  the  face  of 
the  tooth  will  be  nearly  as  long  as  that  found  by  the  first  of  the  arbi- 
trary rules  given  in  (201)  and  the  diameter  of  the  blank  is  deter- 
mined by  simply  adding  to  that  of  the  pitch  circle  2J  times  the 
diametral  pitch. 

On  the  Manufacture  of  Accurate  Gear  Cutters. 

In  cutting  a  spur  wheel,  it  is  obviously  essential  that  the  contour  of 
the  milling  cutter  conform  precisely  to  that  of  the  space  between  two 
adjacent  teeth  :  in  order  to  this  the  process  most  extensively,  and 
until  recently  exclusively,  employed,  involves  :  1st,  the  drawing  of 
the  required  curve  ;  2d,  the  filing  of  a  template  to  that  exact  form  ; 
and  3d,  the  turning  of  the  cutter  to  fit  the  template. 

The  first  improvement  upon  this,  we  believe,  was  the  construction 
by  Messrs.  Brown  &.  Sharpe,  of  Providence,  R.  L,  of  an  ingenious 
piece  of  mechanism  called  an  Epicycloidal  Engine,  by  which  the 
curves  are  traced  automatically  and  with  perfect  precision  upon  the 
template  by  continuous  motion.  There  still  remain  the  two  han el- 
and-eye operations  of  filing  up  the  template,  and  of  turning  the  cutter 
12 


178 


MANUFACTURE    OF   GEAR   CUTTERS. 


to  fit  it  when  done  ;  and  these  operations  are  so  delicate  and  difficult, 
that  the  exact  duplication  of  templates,  cutters  or  wheels  in  this 
manner  is  well  nigh  impossible. 

Yet  this  duplication,  it  is  very  easy  to  see,  is  quite  as  desirable  as  the 
accurate  formation  of  a  single  cutter  ;  and  it  has  been  made  a  matter 
of  perfect  certainty  and  ready  execution  by  means  of  two  machines 
recently  constructed  by  Messrs.  Pratt  &  Whitney,  of  Hartford,  Conn. 
Of  these,  which  are  remarkable  for  the  ingenuity  and  beauty  of 
their  movements,  we  will  now  proceed  to  describe  the  principles  and 
mode  of  action,  though  not  the  exact  details.  The  whole  process 
under  this  system  is  mechanical  and  nearly  automatic ;  a  template 
being — not  traced,  but  milled  out,  by  one  machine,  which  is  subse- 
quently used  in  the  other  as  a  guide  by  which  its  motions  are  so  con- 
trolled as  by  another  milling  operation  to  finish  the  contour  of  tho 
gear  cutter,  whatever  the  size  of  the  tooth  to  be  cut,  to  the  precise 
epicycloidal  form — with  a  minute  and  practically  unimportant  ex- 
ception, as  will  presently  be  explained. 

295,  The  Epicycloidal  Milling  Engine.— In  Fig.  170,  A  is  a  portion 

of  a  flat  ring,  fixed  to  the  fram- 
ing; this  represents  a  pitch 
circle.  B,  is  a  disc,  represent- 
ing the  describing  circle  ;  this 
turns  freely  upon  a  tubular 
stud  E,  fixed  in  the  carrier  C, 
which  turns  about  a  pivot  />, 
fixed  to  the  frame  at  the  centre 
of  A ;  by  means  of  the  clamped 
socket,  capable  of  sliding  upon 
the  rod,  the  position  of  D  may 
be  adjusted  to  suit  the  radius 
of  A.  Thus  as  C  moves,  the 
disc  can  roll  upon  the  edge  of 
A,  and  is  compelled  to  do  so  by 
the  flexible  steel  ribbon  shown 
by  the  heavy  line,  which  is 
wrapped  round  and  secured  to 
both  pieces,  due  allowance  for 
its  thickness  being  made  in  ad- 
justing their  radii.  E'  is  a 
second  tubular  stud  fixed  in  the 
FlG-  17°-  carrier,  at  the  same  distance 

from  the  pitch  circle  as  the  other,  but  on  the  opposite   side  ;  the 


EPICYCLOIDAL   ENGINE. 


179 


centres  of  the  two  studs  lying  on  a  right  line  through  D.  Upon  these 
two  studs  turn  the  two  worm  wheels  F,F',  shown  in  Fig.  171 ;  these 
are  in  a  plane  above  A  and  B,  so  that  the  axis  of  the  worm,  G,  is 
vertically  over  the  common  tangent  of  the  pitch  and  describing  circles  ; 
the  relative  positions  of  these  and  other  parts  will  be  most  clearly  seen 
by  a  study  of  the  vertical  section,  I?ig.  .175.  The  worm  G,  is  sup- 
ported in  bearings  secured  to  the  carrier  (7,  and  is  driven  by  another 
small  worm  turned  by  the  pulley  / ;  the  driving  cord,  passing  through 
suitable  guiding  pulleys,  is  kept  at  a  uniform  tension  by  a  weight,  not 
shown,  however  (7  moves. 
Upon  the  same  studs,  in  a  plane  still  higher  than  the  worm-wheels, 


FIG.  171. 


Fro.  172. 


turn  the  two  discs  ff,  H'9  Figs.  172,  173,  174.  The  diameters  of 
these  are  equal,  and  precisely  the  same  as  those  of  the  describing 
circles  which  they  represent,  with  due  allowance,  again,  for  the  thick- 
ness of  the  steel  ribbon  by  which  these  also  are  connected. 

It  will  be  understood  that  each  of  these  discs  is  secured  to  the 
worm  wheel  below  it,  and  the  outer  one  of  these  to  the  disc  B  ;  so 
that  as  the  worm  G  turns,  H  and  H'  are  rotated  in  opposite  direc- 
tions, the  motion  of  H  being  identical  with  that  of  B  ;  this  last  is  a 
rolling  upon  the  edge  of  A,  the  carrier  C  with  all  its  attached  mechan- 
ism moving  around  D  at  the  same  time.  Ultimately,  then,  the  motions 


180 


EPICYCLOIDAL   ENGINE. 


of  H,  ff',  are  those  of  two  equal  describing  circles  rolling  in  external 
and  internal  contact  with  a  fixed  pitch  circle. 

In  the  edge  of  each  disc  a  semicircular  recess  is  formed,  into  which 
is  accurately  fitted  a  cylinder,  J9  provided  with  flanges,  between  which 
the  discs  fit  so  as  to  prevent  end  play  ;  this  cylinder  is  perforated  for 
the  passage  of  the  steel  ribbon,  the  sides  of  the  opening,  as  shown  in 
Fig.  172,  having  the  same  curvature  as  the  rims  of  the  discs.  Thus 
when  these  recesses  are  opposite  each  other,  as  in  Fig.  173,  the  cylin- 
der /fills  them  both,  and  the  tendency  of  the  steel  ribbon  is  to  carry 
it  along  Avith  H  when.  C  moves  to  one  side  of  this  position,  as  in  Fig. 


FIG.  173. 


FIG.  174. 


174,  and  along  with  IT  when  Amoves  to  the  other  side,  as  in  Fig. 
172. 

This  action  is  made  positively  certain  by  means  of  the  hooks 
Ky  K ',  which  catch  into  recesses  formed  in  the  upper  flange  of 
J,  as  seen  in  Fig.  173.  The  spindles,  with  which  these  hooks  turn, 
extend  through  the  hollow  studs,  and  the  coiled  springs  attached  to 
their  lower  ends,  as  seen  in  Fig.  175,  urge  the  hooks  in  the  directions 
of  their  points  ;  their  motions  being  limited  by  stops  0,  o',  fixed  not 
in  the  discs  H,  H',  but  in  projecting  collars  on  the  upper  ends  of 
the  tubular  studs.  The  action  will  be  readily  traced  by  comparing 


MANUFACTURE   OF   GEAR   CUTTERS. 


181 


Fig.  173  with  Fig.  174 ;  as  C  goes  to  the  left,  the  hook  K'  is  left 
behind,  but  the  other  one,  K9  cannot  escape  from  its  engagement  with 
the  flange  of  / ;  which  accordingly  is  carried  along  with  H  by  the 
combined  action  of  the  hook  and  the  steel  ribbon. 

On  the  top  of  the  upper  flange  of  J  is  secured  a  bracket,  carrying 
the  bearings  of  a  vertical  spindle  Z,  whose  centre  line  is  a  prolongation 
of  that  of  J  itself.  This  spindle  is  driven  by  the  spur  wheel  N,  keyed 
on  its  upper  end,  through  a  flexible  train  of  gearing,  not  shown ;  at 
its  lower  end  it  carries  a  small  milling  cutter,  M,  which  forms  the 
edge  of  the  template,  T,  firmly  clamped  to  the  framing. 

When  the  machine  is  in  operation,  a  heavy  weight,  not  shown,  acts 
to  move  C  about  the  pivot  Z),  being  attached  to  the  carrier  by  a  cord 
guided  by  suitably  arranged  pulleys  ;  this  keeps  the  cutter  M  up  to 


Fia.  175. 

its  work,  while  the  spindle  L  is  independently  driven,  and  the  duty 
left  for  the  worm  G  to  perform  is  merely  that  of  controlling  the 
motions  of  the  cutter  by  the  means  above  described,  and  regulating 
their  speed. 

The  centre  line  of  the  cutter  is  thus  automatically  compelled  to 
travel  in  the  path  RS,  composed  of  an  epicycloid  and  a  hypocycloid, 
if  A  be  a  segment  of  a  circle  as  here  shown ;  or  of  two  cycloids,  if  A 
be  a  straight  bar.  The  radius  of  the  cutter  being  constant,  the  edge 
of  the  template  T  is  cut  to  an  outline  also  composed  of  two  curves ; 
since  the  radius  of  J/is  small,  this  outline  closely  resembles  RS;  but 
particular  attention  is  called  to  the  fact  that  it  is  not  identical  with 
it,  nor  yet  composed  of  truly  epicycloidal  curves  of  any  generation 
ivhatever  ;  the  result  of  which  will  be  subsequently  explained. 

The  diameter  of  the  discs  which   act  as  describing  circles  is  7£ 


182 


PANTAGRAPHIC  CUTTER  ENGINE. 


inches,  and  that  of  the  milling  cutter,  which  shapes  the  edge  of  the 
template,  is  J  of  an  inch. 

Now  if  we  make  a  set  of  1-pitch  wheels  with  these  describing  circles, 
the  one  with  fifteen  teeth  will  have  radial  flanks.  The  curves  will  be  the 
same  whatever  the  pitch  ;  but,  as  shown  in  Fig.  176,  the  blank  should  be 
adjusted  in  the  epicycloidal  engine  so  that  its  lower  edge  shall  be  TV 
of  an  inch  (the  radius  of  the  cutter  M )  above  the  bottom  of  the  space  ; 
also  its  relation  to  the  side  of  the  proposed  tooth  should  be  as  here 
shown.  And,  as  previously  explained,  the  depth  of  the  space  depends 
upon  the  pitch.  In  the  system  adopted  by  the  Pratt  &  Whitney  Com- 
pany the  whole  height  of  the  tooth  is  2J  times  the  diametral  pitch,  the 
projection  outside  the  pitch  circle  being  just  equal  to  the  pitch,  so  that 
diameter  of  blank  =  diameter  of  pitch  circle  +  2  x  diametral  pitch. 

We  have  now  to  show  how,  from  a  single  set  of  what  may  be  called 
1-pitch  templates,  complete-  sets  of  cutters  of  the  true  epicycloidal 
contour  may  be  made  of  the  same  or  any  less  pitch. 

The  Pantagraphic  Cutter  Engine. 

296.  In  Fig.  176,  the  edge  TT,  is  shaped  by  the  cutter  M,  whose  cen- 
tre travels  in  the  path  RS,  therefore  these  two  lines  are  at  a  constant 


FIG.  176.  FIG.  177. 

normal  distance  from  each  other.  Let  a  roller  P,  of  any  reasonable 
diameter,  be  run  along  TT\  its  centre  will  trace  the  line  UV,  which 
is  at  a  constant  normal  distance  from  TT,  and  therefore  from  US. 
Let  the  normal  distance  between  ?7Fand  ES  be  the  radius  of  another 
milling  cutter,  N,  having  the  same  axis  as  the  roller  P,  and  carried 


MANUFACTURE   OF   GEAR   CUTTERS.  183 

by  it,  but  in  a  different  plane,  as  shown  in  the  side  view ;  then  what- 
ever .2V  cuts  will  have  It/S  for  its  contour  if  it  lie  upon  the  same  side 
of  the  cutter  as  the  template. 

Now  if  TT  be  a  1-pitch  template  above  mentioned,  it  is  clear  that 
N  will  correctly  shape  a  cutting  edge  of  a  gear  cutter  for  a  1-pitch 
wheel.  The  same  figure,  reduced  to  half  size,  would  correctly  repre- 
sent the  formation  of  a  cutter  for  a  2-piteh  wheel  of  the  same  number 
of  teeth  ;  if  to  quarter  size,  that  of  a  cutter  for  a  4-pitch  wheel,  and 
so  on. 

But  since  the  actual  size  and  curvature  of  the  contour  thus  deter- 
mined depend  upon  the  dimensions  and  motion  of  the  cutter  N,  it 
will  be  seen  that  the  same  result  will  practically  be  accomplished  if 
these  only  be  reduced  ;  the  size  of  the  template,  the  diameter  and  the 
path  of  the  roller  remaining  unchanged. 

The  nature  of  the  means  by  which  this  is  effected  in  the  Panta- 
graphic  Cutter  Engine  is  illustrated  in  Fig.  177.  The  milling  cutter, 
N9  is  driven  by  a  flexible  train  acting  upon  the  wheel,  0 ;  its  spindle 
is  carried  by  the  bracket,  B,  which  can  slide  from  right  to  left  upon 
the  piece,  A>  and  this,  again,  is  free  to  slide  in  the  frame  F.  These 
two  motions  are  in  horizontal  planes,  and  perpendicular  to  each  other. 

The  upper  end  of  the  long  lever,  PC,  is  formed  into  a  ball,  working 
in  a  socket  which  is  fixed  to  B.  Over  the  cylindrical  upper  part  of 
this  lever  slides  an  accurately  fitted  sleeve,  D,  partly  spherical  exter- 
nally, and  working  in  a  socket  which  can  be  clamped  at  any  height 
on  the  frame  F.  The  lower  end,  P,  of  this  lever  being  accurately 
turned,  corresponds  to  the  roller,  P,  in  Fig.  176,  and  is  moved  along 
the  edge  of  the  template,  T,  which  is  fastened  in  the  frame  in  an  inva- 
riable position. 

By  clamping  D  at  various  heights,  the  ratio  of  the  lever  arms,  TD, 
DC,  may  be  varied  at  will,  and  the  axis  of  JVmade  to  travel  in  a  path 
similar  to  that  of  the  axis  of  P,  but  as  many  times  smaller  as  we 
choose;  and  the  diameter  of  Nis  made  less  than  that  of  P  in  the 
same  proportion. 

The  template  being  on  the  left  of  the  roller,  the  cutter  to  be  shaped 
is  placed  on  the  right  of  N,  as  shown  in  the  plan  view  at  Z,  because 
the  lever  reverses  the  movement. 

This  arrangement  is  not  mathematically  perfect,  by  reason  of  the 
angular  motion  of  the  lever.  This  is,  however,  very  small,  owing  to 
the  length  of  the  lever  ;  it  might  have  been  compensated  for  by  the 
introduction  of  another  universal  joint,  which  would  practically  have 
introduced  an  error  greater  than  the  one  to  be  obviated,  and  it  has 
with  good  judgment  been  omitted. 


184 


MANUFACTURE   OF   GEAR   CUTTERS. 


The  gear  cutter  is  turned  nearly  to  the  required  form,  the  notches 
are  cut  in  it,  and  the  duty  of  the  pantagraphic  engine  is  merely  to 
put  the  finishing  touch  to  each  cutting  edge  and  give  it  the  correct 
outline.  It  is  obvious  that  this  machine  is  in  no  way  connected  with, 
or  dependent  upon,  the  epicycloidal  engine  ;  but  by  the  use  of  proper 
templates  it  will  make  cutters  for  any  desired  form  of  tooth  ;  and  by 
its  aid  exact  duplicates  may  be  made  in  any  numbers  with  the  greatest 
facility. 

TJieoretical  Defects  of  the  System. 

297.  It  forms  no  part  of  our  plan  to  represent  as  perfect  that  which 
is  not  so.  And  there  are  one  or  two  facts  which  at  first  thought 
might  seem  serious  objections  to  the  adoption  of  the  epicycloidal  sys- 
tem. These  are  : 

1.  It  is  physically  impossible  to  mill  out  a  concave  cycloid,  by  any 


FIGS.  178  and  179, 

means  whatever,  because  at  the  pitch  line  its  radius  of  curvature  is 
zero,  and  a  milling  cutter  must  have  a  sensible  diameter. 

2.  It  is  impossible  to  mill  out  even  a  convex  cycloid  or  epicycloid,  by 
the  means  and  in  the  manner  above  described. 

This  is  on  account  of  a  hitherto  unnoticed  peculiarity  of  the  curve 
at  a  constant  normal  distance  from  the  cycloid.  In  order  to  show 
this  clearly,  we  have,  in  Fig.  178,  enormously  exaggerated  the  radius, 
CD,  of  the  milling  cutter  (M  of  Figs.  174  and  175).  The  outer  curve, 
HL,  evidently  could  be  milled  out  by  the  cutter,  whose  centre  travels 


MINUTENESS   OF   EKROR.  185 

in  the  cycloid  CA  ;  it  resembles  the  cycloid  somewhat  in  form,  and 
presents  no  remarkable  features.  But  the  inner  one  is  quite  different ; 
it  starts  at  D,  and  at  first  goes  down,  inside  the  circle  whose  radius  is 
CD,  forms  a  cusp  at  E,  then  begins  to  rise,  crossing  this  circle  at  G, 
and  the  base  line  at  F.  It  will  be  seen  then  that  if  the  centre  of  the 
cutter  travel  in  the  cycloid  A  C,  its  edge  will  cut  away  the  part  GED, 
leaving  the  template  of  the  form  OGI.  Now  if  a  roller  of  the  same 
radius  CD,  he  rolled  along  this  edge,  its  centre  will  travel  in  the  cy- 
cloid from  A,  to  the  point  P,  where  a  normal  from  G  cuts  it  ;  then 
the  roller  will  turn  upon  G  as  a  fulcrum,  and  its  centre  will  travel 
from  P  to  (7,  in  a  circular  arc,  whose  radius  is  GP  =  CD. 

That  is  to  say,  even  a  roller  of  the  same  size  as  the  original  milling 
cutter,  will  not  retrace  completely  the  cycloidai  path  in  which  the 
cutter  traveled. 

Now  in  making  a  rack  template,  the  cutter,  after  reaching  C, 
travels  in  the  reversed  cycloid  CR,  its  left-hand  edge,  therefore,  mill- 
ing out  a  curve  DK,  similar  to  HL.  This  curve  lies  wholly  outside 
the  circle/)/,  and  therefore  cuts  OG  at  a  point  between  F  and  G, 
very  near  to  G.  This  point  of  intersection  is  marked  S  in  Fig.' 
179,  where  the  actual  form  of  the  template  OSK  is  shown.  The 
roller  which  is  run  along  this  template  is  larger,  as  has  been  explained, 
than  the  milling  cutter.  When  the  point  of  contact  reaches  S  (which 
is  so  near  to  G  that  they  practically  coincide),  this  roller  cannot  now 
swing  about  S  through  an  angle  so  great  as  PGCof  Fig.  12  ;  because 
at  the  root  D,  the  radius  of  curvature  of  /)A^is  only  equal  to  that  of 
the  cutter,  and  G  and  S  are  so  near  the  root  that  the  curvature  of  SK, 
near  the  latter  point,  is  greater  than  that  of  the  roller.  Consequently 
there  must  be  some  point  U  in  the  path  of  the  centre  of  the  roller, 
such  that,  when  the  centre  reaches  it,  the  circumference  will  pass 
through  S,  and  be  also  tangent  to  SK.  Let  T  be  the  point  of  tan- 
gency  ;  draw  $Z7and  TU9  cutting  the  cycloidai  path  AR  in  X  and 
Y.  Then,  UY  being  the  radius  of  the  new  milling  cutter  (corre- 
sponding to  JVof  Fig.  9),  it  is  clear  that  in  the  outline  of  the  gear- 
cutter  shaped  by  it,  the  circular  arc  XY  will  be  substituted  for  the 
true  cycloid. 

The  System.  Practically  Perfect. 

298.  The  above  defects  undeniably  exist ;  now,  what  do  they  amount 
to  ?  The  diagrams,  Figs.  178  and  179,  are  drawn  purposely  with 
these  sources  of  error  greatly  exaggerated  in  order  to  make  their  nature 
apparent  and  their  existence  sensible.  The  diameters  used  in  prac- 
tice, as  previously  stated,  are  :  describing  circle,  7£  inches  ;  cutter 


186  EQUIDISTANT   CUTTEES. 

for  shaping  template,  J  of  an  inch  ;  roller  used  against  edge  of  tem- 
plate, 1J  inches  ;  cutter  for  shaping  a  1-pitch  gear-cutter,  1  inch. 

With  these  data  the  author  has  found  that  the  total  length  of  the 
arc  XY  of  Fig.  179,  which  appears  instead  of  the  cycloid  in  the  out- 
line of  a  cutter  for  a  1-pitch  rack,  is  less  than  0.0175  inch  ;  the  real 
deviation  from  the  true  form,  obviously,  must  be  much  less  than  that. 
It  need  hardly  be  stated  that  the  effect  upon  the  velocity  ratio  of  an 
error  so  minute,  and  in  that  part  of  the  contour,  is  so  extremely  small 
as  to  defy  detection.  And  the  best  proof  of  the  practical  perfection 
of  this  system  of  making  epicycloidal  teeth  is  found  in  the  smoothness 
and  precision  with  which  the  wheels  run,  in  which  respects  they  have 
proved,  after  some  years  of  trial,  to  be  unsurpassed.  And  we  repeat, 
that  objection  taken,  on  whatever  grounds,  to  the  epicycloidal  form 
of  tooth,  has  no  bearing  upon  the  method  above  described  of  produc- 
ing duplicate  cutters  for  teeth  of  any  form,  which  the  pantagraphic 
engine  will  make  with  the  same  facility  and  exactness,  if  furnished 
with  the  proper  templates. 

On  the  Determination  of  a  Series  of  Cutters. 

299.  In  making  a  set  of  interchangeable  wheels,  upon  any  system 
whatever,  every  additional  tooth  changes  the  diameter  of  tho  wheel 
and  the  form  of  the  acting  curves,  so  that  in  order  to  secure  absolute 
theoretical  accuracy,  it  would  be  necessary  to  have  as  many  different 
cutters  as  there  are  wheels.  This  is  clearly  out  of  the  question,  and 
fortunately,  the  proportional  increment,  and  the  actual  change  of 
form,  become  less  as  the  wheel  becomes  larger;  and  this  alteration 
in  the  outline  soon  becomes  imperceptible.  Going  still  farther,  we 
can  presently  add  more  than  one  tooth  without  producing  a  sensible 
variation  in  the  contour.  That  is  to  say,  several  wheels  can  be  cut 
with  the  same  cutter,  without  introducing  a  perceptible  error.  Pro- 
ceeding in  this  way  we  reach  the  conclusion,  that  instead  of  an  infi- 
nite number  of  cutters,  a  quite  limited  number  will  suffice  to  cut  all 
the  wheels  of  such  a  set,  from  the  smallest  up  to  a  rack,  with  a  suffi- 
cient degree  of  accuracy  for  all  ordinary  purposes.  We  have  then  to 
decide  how  many  cutters  shall  be  made,  and  also  for  what  numbers  of 
teeth  these  should  be  specially  adapted.  The  answer  to  the  first 
question  evidently  depends  upon  how  great  a  deviation  from  absolute 
accuracy  is  considered  admissible.  In  regard  to  the  second,  it  appears 
reasonable,  not  to  say  axiomatic,  that  errors  should  be  uniformly 
distributed  through  the  series,  and  that  this  will  be  effected  by  mak- 
ing the  greatest  difference  in  form  the  same  between  any  two  consec- 
utive cutters. 


WILLIS'S  SEEIES — GRAFT'S  FORMULA. 


187 


300.  In  relation  to  this,  Professor  Willis  says  : 

«  _  _it  appeared  worth  while  to  investigate  some  rule  by  which 
the  necessary  cutters  could  be  determined  for  a  set  of  wheels,  so  as  to 
incur  the  least  possible  chance  of  error.  To  this  effect 
I  calculated,  by  a  method  sufficiently  accurate  for  the 
purpose,  the  following  series  of  what  may  be  termed 
equidistant  values  of  cutters  ;  that  is,  a  table  of  cut- 
ters so  arranged,  that  the  same  difference  of  form  ex- 
ists between  any  two  consecutive  numbers."  He  gives 
us  no  clue  to  the  method  by  which  these  numbers  were 
found,  nor  any  explanation  of  the  sign  x ,  which  ap- 
pears in  two  places.  The  teeth  are  epicycloidal  (the 
describing  circle  being  such  as  to  give  radial  flanks  to 
the  wheel  of  12  teeth),  and  have  "  the  usual  adden- 
dum " — of  which  the  exact  amount  is  not  stated.  We 
here  insert  the  figures  as  he  gives  them,*  only  num- 
bered in  the  reverse  order ;  and  they  appear  to  have 
been  extensively  adopted  in  practice,  and  that  too 
under  varying  conditions,  which,  as  will  presently  be 
shown  would  affect  the  values  of  the  terms  of  a  truly 
equidistant  series. 

301.  The  only  rule  or  formula  for  computing  such 
a  series  that  we  have  met  with,  is  given  by  Mr.  G.  B. 
Grant,  as  follows  : 

an 


z  =^ 
n  = 


as 

n  —  s  H • 

z 

in  which 

a  =  the  number  of  teeth  on  the  least  wheel,  usually  12  ; 

"       "     "      "    "  greatest  "         "      «=> ; 

"  "  cutters  in  a  series  to  cut  from  a  to  z ; 
s  —  "  "  in  the  series,  of  any  particular  cutter  ; 
t  =  "  "  of  teeth  on  the  last  wheel  to  be  cut  by  s. 
This  formula  is  based  upon  the  following  hypothe- 
sis :  The  pitch  arc  is  greater  upon  the  outside  of  the 
wheel  than  on  the  pitch  circle,  by  an  amount  which 
in  wheels  of  equal  pitch  and  constant  addendum,  de- 
pends upon  the  number  of  teeth ;  the  smaller  the 
wheel,  the  greater  is  the  difference,  and  vice  versv, 
the  arcs  being  equal  in  the  case  of  a  rack.  Now  selecting  any  two 


1 

12 

2 

x 

3 

13 

4 

14 

5 

.15 

6 

x 

7 

16 

8 

17 

9 

19 

10 

20 

11 

21 

12 

23 

13 

25 

14 

27 

15 

30 

16 

34 

17 

38  . 

18 

43 

19 

50 

20 

60 

21 

76 

22 

100 

23 

150 

24 

300 

25 

00 

Principles  of  Mechanism,"  p.  141. 


188 


EQUIDISTANT   CUTTERS. 


wheels  as  the  least  and  greatest  of  a  series,  it  is  possible  to  interpolate 
others  whose  numbers  of  teeth  are  such  that  this  difference,  between 
the  pitch  arc  as  measured  on  the  pitch  circle  and  on  the  extreme 
perimeter,  shall  be  the  same  for  each  :  and  these  numbers  are  the 
values  of  t  as  found  by  Mr.  Grant's  formula,  in  which  it  is  assumed, 
although  upon  what  ground  is  not  apparent,  that  the  consecutive 
wheels  of  such  a  series  will  differ  equally  throughout  in  respect  to  the 
outline  of  the  tooth.  The  proper  location  of  each  cutter  in  its  own 
interval,  is  determined  by  doubling  the  number  of  terms  in  the  series, 
and  taking  each  alternate  value  of  t  as  the  number  of  teeth  for  which 
the  preceding  cutter  is  to  be  made  exactly  correct. 
It  may  be  admitted  that  tolerably  close  approximations,  or  even 


FIG.  180. 


Fio.  181. 


correct  results,  may  be  obtained  by  this  formula  under  some  condi- 
tions. But  it  contains  no  internal  evidence  as  to  when  or  whether 
this  will  be  the  case — -it  is  independent  of  all  the  elements  of  the 
problem  which  affect  the  form  or  the  length  of  the  tooth,  and  should, 
therefore,  if  true  at  all,  be  applicable  to  all  systems  and  under  all 
circumstances.  "Which  would  be  very  convenient,  unquestionably ; 
but  these  elements  cannot  be  eliminated  without  vitiating  the  results 
and  destroying  the  "equidistant"  characteristic  of  the  series. 

302.  This  is  best  illustrated  by  considering  the  question  in  relation 
to  the  epicycloidal  system,  which,  at  least,  admits  of  a  direct  and 
exact  solution.  In  Fig.  180,  let  C  be  the  centre  of  the  describing 
circle,  VW  an  arc  of  the  pitch  circle  of  the  smallest  wheel,  and  D 


EPICYCLOIDAL   SYSTEM.  189 

its  centre  ;  also  let  AF\>Q  the  addendum.  Then  a  circle  through  F 
about  centre  D  determines  the  arc  AB  of  the  describing  circle,  which 
by  rolling  upon  an  equal  arc  AH  of  the  pitch  circle  traces  the  epicy- 
cloidal  face  BH  si  the  tooth  for  that  wheel.  Again,  FE  perpendic- 
ular to  CD  cuts  off  an  arc  AE,  which  by  rolling  upon  the  tangent  at 
A  describes  the  cycloidal  face  EG  for  the  tooth  of  a  rack. 

Extending  the  tangent  line  toward  the  right,  set  off  upon  it  AT  = 
arc  A M  =  -J-  space,  and  draw  IL,  MP,  respectively  similar  and  equal 
to  GE,  HB.  Then  regarding  CD  as  the  central  plane  of  a  series  of 
cutters,  it  will  be  seen,  by  comparing  this  figure  with  the  following 
one,  which  represents  a  section  of  a  milling  cutter  at  work,  that  MP 
will  be  the  contour  of  the  base  of  the  one  for  the  smallest  wheel,  and 
IL  the  corresponding  line  of  the  cutter  for  the  rack. 

Proceeding  in  a  similar  manner  with  other  pitch  circles,  it  is  ob- 
yious  that  the  faces  for  all  intermediate  wheels  will  lie  between  MP 
and  IL,  their  highest  points  lying  in  a  curve  LP ; 
which  is  shown  on  a  larger  scale  in  Fig.  182.  Now 
these  epicycloids  diverge  and  differ  in  form  from  each 
other  the  more  the  farther  they  extend  from  the 
pitch  circles  ;  if  then  they  be  so  selected  as  in  this 
figure  that  they  divide  the  locus  ZPinto  equal  parts, 
the  greatest  variation  of  each  from  the  one  next  in 
order  will  be  constant  for  the  whole  series. 

And  this  is  sufficient  for  our  purpose,  it  being  self-evident  that  the 
errors  of  which  the  distribution  is  to  be  equalized,  should  be  taken 
at  the  maximum  ;  and  again,  it  is  not  necessary  to  take  into  consid- 
eration the  flanks  of  the  teeth,  to  which  the  like  method  is  equally 
applicable,  because  they  vary  from  each  other  to  a  much  less  extent 
than  the  faces. 

303.  We  have,  then,  a  key  to  the  perfect  solution  of  the  problem. 
The  first  step  is  to  determine  LP ;  this  is  done  by  assuming  various 
pitch  circles,  and  constructing  the  face  for  a  tooth  of  each,  arranging 
these  with  reference  to  the  point  A  as  above  described,  until  a  suffi- 
cient number  of  points  have  been  located  to  enable  the  curve  to  be 
properly  developed  and  drawn.  Having  decided  upon  the  number  of 
cutters  in  the  proposed  series,  LP  is  to  be  divided  into  a  correspond- 
ing number  of  equal  parts.  In  regard  to  the  practical  execution  of 
this  process,  it  may  be  remarked  that  although  the  curve  LP  may  be 
of  a  complicated  nature,  since  it  has  a  sort  of  transcendental  depend- 
ence upon  the  epicycloid,  which  is  itself  transcendental,  yet  in  the 
portion  of  it  with  which  we  have  to  deal  in  connection  with  wheels 
ranging  from  twelve  teeth  up  to  a  rack,  its  curvature  changes  so 


190  EQUIDISTAKT   CUTTERS. 

slightly  that  it  may  without  causing  any  appreciable  error  be  consid- 
ered as  a  circular  arc. 

We  have  next  to  find  the  radius  of  the  wheel,  whose  tooth  would 
fall  at  each  point  of  subdivision  ;  this  is  readily  done  as  shown  in  Fig. 
180,  thus :  let  S  be  one  of  these  points  upon  LP,  we  have  but  to 
draw  FS,  and  bisect  it  by  a  perpendicular,  which  latter  line  will  cut 
CD  at  0,  the  centre  of  the  required  wheel. 

This  determines  the  limits  between  which  each  cutter  is  to  be  used  : 
thus,  if  in  Fig.  182  we  suppose  a  series  of  four  cutters  to  be  required, 
PM  being  the  face  of  the  smallest  wheel  and  LI  that  of  the  largest, 
then  the  first  cutter  must  cut  from  P  to  Q,  the  next  from  Q  to  S, 
and  so  on.  Evidently,  each  cutter  should  be  made  exactly  right  for 
a  wheel  whose  tooth  would  fall  at  the  point  bisecting  the  arc  of  LP 
which  limits  its  range  of  action  ;  so  that  the  location  of  the  first  is 
f ound  by  dividing  PQ  in  half,  and  determining  as  above  the  radius 
of  the  wheel  corresponding  to  this  point  of  division. 

304.  The  same  general  mode  of  proceeding  may  be  employed  with 
teeth  of  any  form,  the  operation  of  determining  points  in  the  locus 
LP  differing  in  detail  according  to  the  peculiarities  of  each  system 
of  gearing.     It  is  apparent  that  this  process  requires  us  to  take  into 
account  in  every  case  not  only  the  addendum,  but  the  path  of  con- 
tact.    The  fact  that  the  latter  is  in  the  epicycloidal  system  identical 
with  the  describing  circle,  renders  comparatively  simple  what  in  other 
systems  might  be  very  complicated,  and  enables  us  by  ordinary  trigo- 
nometrical operations  to  locate  any  required  number  of  points  in  LP 
with  great  accuracy. 

And  pursuing  this  line  of  investigation,  we  find  that  the  curvature,f 
position  and  magnitude  of  LP  are  affected  by  changes  in  either 
the  length  of  the  tooth  or  in  the  diameter  of  the  describing  circle,  in 
such  a  manner  and  to  such  an  extent  as  sensibly  to  affect  in  turn  at 
least  the  higher  values  of  a  series  of  equidistant  cutters. 

Now  it  costs  no  more  to  make  or  to  use  a  correct  series  than  an  in- 
correct one  ;  and  in  adopting  the  epicycloidal  system,  there  is  no 
excuse  for  resting  contented  with  approximations,  close  or  other- 
wise, since  by  the  process  above  outlined,  results  may  be  obtained 
which  are  exact  to  a  single  tooth.  On  this  account,  and  also  because 
this  system  has  been  and  is  likely  to  be  more  extensively  employed 
than  any  other,  we  have  made  the  required  determinations  with  great 
care,  in  order  to  reduce  the  magnitude,  and  correct  the  unequal  dis- 
tribution, of  errors  in  the  series  in  use. 

305.  First,  taking  the  diametral  pitch  as  unity,  with  a  describing 
circle  with  diameter  —  7i,  and  addendum  =  1,  the  locus  LP  was 


e 
^s. 


FIG.  183. 


EPICYCLOIDAL   SYSTEM.  191 

accurately  determined  by  calculating  the  positions  of  a  great  number 

of  points.     This  curve  is  shown,  greatly  enlarged,  in  Fig.  183,  LN 

in  this  figure  corresponding  to  the  same  line  in  Fig.  180,  being  the 

perpendicular  let  fall 

from   L    upon    PR. 

The    numbers    upon 

the   horizontal    ordi- 

nates     indicate     the 

diameters  of  the  pitch 

circles    assumed    for 

the  computations,  or, 

what     is    the    same 

thing,  the  numbers  of 

teeth. 

Next  taking  any  point  thus  determined,  at  pleasure,  and  assuming 
it  to  lie  upon  a  circle  passing  through  L  and  P,  the  radius  of  that 
circle  was  calculated — which  being  repeated  for  various  other  points, 
the  close  agreement  of  the  radii  proved  that  for  the  purpose  in  view 
the  curve  might  be  regarded  as  a  circular  arc.  Upon  this  assumption, 
the  location  of  any  point  of  subdivision,  with  reference  to  CD  and  the 
point  F  of  Fig.  180,  is  known,  and  the  number  of  teeth  upon  the 
wheel  from  which  that  point  would  have  been  derived,  can  be  calcu- 
lated trigonometrically  ;  this  converse  process  is  quite  different  from 
that  by  which  a  point  upon  the  curve  is  found,  so  that  the  concordance 
of  the  results  affords  a  rigid  test  of  the  accuracy  both  of  the  work  and 
of  the  whole  method. 

It  was  not  to  be  expected  that  the  points  of  exact  subdivision  of 
this  curve  should  correspond  precisely  to  whole  numbers  of  teeth  ;  on 
the  contrary,  fractional  terms  were  to  be  looked  for  as  a  matter  of 
course,  but  since  the  series  must  in  practice  consist  of  whole  numbers 
only,  the  nearest  integer  was  taken  in  each  case. 

306.  It  will  be  observed  that  the  middle  point  of  LP  lies  very  close 
to  the  ordinate  for  the  wheel  of  24  teeth.  Therefore,  in  a  set  of  24 
cutters,  a  separate  one  should  be  made  for  each  wheel  from  12  to  23 
teeth  inclusive,  thus  confining  both  the  errors  and  the  calculations  to 
the  12  subdivisions  of  the  upper  half  of  the  curve. 

A  series  thus  determined  is  given  in  the  first  column  of  the  com- 
parative table  following :  in  regard  to  which  it  should  be  stated  that 
the  24th  cutter  is  practically  used  only  for  very  large  wheels,  an  extra 
one  being  added  for  the  rack. 

When  this  describing  circle  of  74-  was  first  introduced,  a  series  of 
cutters  was  employed,  the  values  for  which  are  given  in  the  second 


192 


EQUIDISTANT   CUTTERS. 


column  ;  these  figures  were  copied  from  those  of  Prof.  Willis,  with 
the  omission  of  his  unintelligible  sign  x ,  and  the  interpolation  of  a 
cutter  for  18  teeth. 

In  both  cases,  the  cutters  used  for  only  one  wheel  each  are  correct ; 
and  the  others  being  "  equidistant  "  in  the  first  series,  we  have  for  the 
purpose  of  a  graphical  comparison  set  up  in  Fig.  184,  at  equal  inter- 
vals, an  ordinate  for 
each     cutter,     of     a 
length      proportional 
to     the     number    of 
teeth  upon  the  small- 
est wheel  for  which  it 
is  to  be  used,  as  given 
in  the  first   column.; 
the  curve  A  A  thus  de- 
termined,   then,    rep- 
resents the  true  series. 
^10<  184t  The  intention  was  to 

make  the  second  series  also  "equidistant,"  and  witli  the  same  differ- 
ence between  consecutive  numbers  as  in  the  first ;  accordingly  the 
curve  BB  is  constructed  by  setting  off  on  the  same  ordinates  the  cor- 
responding numbers  in  the  second  column  ;  and  the  discrepancy 
between  these  curves  exhibits  very  clearly  the  inequality  in  the  dis- 
tribution of  the  errors  in  this  series,  the  use  of  which  was  promptly 
abandoned  as  soon  as  this  was  pointed  out. 

In  the  third  column  we  have  a  series  computed  by  Mr.  Grant's 
formula,  applied  to  determine  the  values  for  12  cutters  to  cut  from 
24  teeth  up  to  a  rack,  the  first  12  being  made  exact,  as  in  the  other 
cases.  This  formula,  as  already  stated,  would  give  the  same  results 
for  any  describing  circle  and  any  addendum  ;  but  changes  in  these 
will  in  fact  sensibly  affect  the  values  of  the  terms  in  this  series.  For 
example,  Prof.  Willis  made  use  of  a  describing  circle  whose  diameter 
is  half  that  of  the  wheel  of  12  teeth  ;  calling  this  diameter,  then,  6, 
and  the  addendum  unity  as  before,  we  find  the  curve  LP  to  be  differ- 
ent from  that  corresponding  to  the  larger  describing  circle  before 
used.  Its  middle  point,  however,  still  lies  so  close  to  the  ordinate 
for  24  teeth,  that  the  upper  half  of  the  curve  only  need  be  dealt  with 
in  computation,  and  the  correct  series  with  these  data  is  given  in  the 
fourth  column.  Again,  taking  the  original  describing  circle  of  di- 
ameter =  7J,  but  reducing  the  addendum  to  |,  the  values  for  the 
series  of  24  cutters  appear  as  given  in  column  fifth.  By  comparing 
this  with  the  first  column,  it  will  be  seen  that  the  higher  values  only 


EFFECT   OF   VARYING    CONDITIONS. 


193 


are  affected  by  this  reduction  of  the  addendum ;  and  it  is  important, 
since  a  series  once  adopted  is  in  the  nature  of  things  inflexible,  and 
yet  it  may  at  times  be  necessary  to  reduce  the  addendum  for  the  pur- 
pose of  varying  the  amount  of  approaching  or  receding  action,  to  note 
that  even  these  values  are  affected  in  a  far  less  proportion  than  that 
in  which  the  difference  between  the  consecutive  cutters  is  reduced. 

This  will  be  clearly  seen  by  the  aid  of  Fig.  185,  in  which  the  curves 
0,  B,  C,  represent  the  locus  LP  of  Fig.  180,  for  the  first,  fourth  and 
fifth  columns  respectively  ;  o,  #,  and  c  being  the  corresponding  centres 
of  curvature. 

Taking  the  twenty-fourth  part  of  each  of  these  curves  as  the 
greatest  difference  in  form  between  the  consecutive  cutters  of  the 
series  to  which  it  belongs,  we  have  the  following  values  ; 


Linear  Variation. 


I.  0.0156645. 

IV.  0.0158606. 

V.  0.0046068. 


VI.   0.0147569. 


307,  We  do  not  consider  it  worth  while  to  discuss  the  determination 
of  any  series  consisting  of  a  less  number  of  cutters  than  24,  because 
the  tendency  of  modern  practice  is  for- 
tunately more  and  more  toward  the  at- 
tainment of  the  greatest  accuracy  con- 
sistent with  a  reasonable  expenditure  ; 
aud  this  would  call  for  an  increase  rather 
than  a  diminution  of  the  numbers.  We 
therefore,  give,  finally,  in  the  sixth 
column  of  the  table,  a  series  of  27  cut- 
ters especially  constructed  for  use  in 
connection  with  the  epicycloidal  and 
pantagraphic  engines  above  described. 
Of  these,  14  are  exactly  right,  there 
being  one  for  each  wheel  from  12  to  25 
teeth  inclusive  ;  the  next  12  are  com- 
puted to  cut  from  26  teeth  up  to  a 
rack;  but  the  26th  is  used  only  for  FIG.  iss. 

wheels  having  322  teeth  or  upward,  an  extra  cutter  being  added  for 
the  rack. 

Much  smaller  series  have  been  employed,  some  makers  contenting 
themselves  with  even  so  few  as  eight  cutters  for  all  pitches  ;  this,  as 
an  examination  of  Fig.  183  will  show  without  further  argument,  can 
hardly  be  said  to  give  a  reasonable  approach  to  accuracy,  especially 
in  the  case  of  wheels  with  comparatively  few  teeth.  Others  again 
13 


194 


VARIOUS   SERIES   COMPARED. 


1  g 
*  1 

NUMBERS   OF  TEETH   CUT  BY  EACH   COTTER. 

I. 

II. 

III. 

IV. 

V. 

VI. 

1 

12 

12 

12 

12 

12 

12 

2 

13 

13 

13 

13 

13 

13 

3 

14 

14 

14 

14 

14 

14 

4 

15 

15 

15 

15 

15 

15 

5 

16 

16 

16 

16 

16 

16 

6 

17 

17 

17 

17 

17 

17 

7 

18 

18 

18 

18 

18 

18 

8 

19 

19 

19 

19 

19 

19 

9 

20 

20 

20 

20 

20 

20 

10 

.      21 

21-22 

21 

21 

21 

21 

11 

22 

23-24 

22 

22 

22 

22 

12 

23 

25-26 

.  23 

23 

23 

23 

13 

24-25 

27-29 

24  26 

24-26 

24-25 

24 

14 

26-28 

30-33 

27-29 

27-29 

26-28 

25 

15 

29-32 

34-37 

30-32 

30-32 

29-32 

26-27 

16 

33-36 

38-42 

83-36 

33-36 

33-36 

28-30 

17 

37-41 

43-49 

37-41 

37-41 

37-41 

31-34 

18 

42-49 

50-59 

42-48 

42-48 

42-48 

35-38 

19 

50-59 

60-75 

49-58 

49-58 

49-58 

39-44 

20 

60-74 

76-99 

59-72 

59-73 

59-81 

45-52 

21 

75-99 

100-149 

78-96 

74-97 

82-109 

53-63 

22 

100-150 

150-299 

97-144 

98-147 

110-165 

64-79 

23 

151-302 

300-co 

145-288 

148-296 

166-334 

80-106 

24 

303-oo 

oo 

289-oo 

297-co 

335-co 

107-159 

25 

00 

oo 

00 

00 

160-321 

26 

322-00 

27 

00 

have  used  eight  for  small  pitches,  increasing  the  number  of  cutters 
as  the  pitch  of  the  tooth  increases.  This  practice  seems  to  be  based 
on  the  idea  that  the  actual  amount  of  error  in  the  form  of  the  cutter 
only  is  to  be  taken  into  account ;  which  may  be  tenable  if  the  forma- 
tion of  the  templates  and  cutters  be  dependent  upon  hand-and-eye  op- 
erations, but  not  otherwise  ;  since  the  proportionate  error,  and  the  effect 
upon  the  velocity  ratio,  will  be  the  same  whatever  the  size  of  the  tooth. 


CHAPTER  XL 


1.  TWISTED  SPUR  GEARING.— Hooke's  Stepped  Wheels.     Twisted  Wheels.    Rota- 

tion not  due  to  Screw-like  Action.  Elimination  of  Sliding.  Neutralization 
of  End  Pressure. 

2.  PIN  GEARING. — Generation  of    Elementary    Tooth.     Derivation  of  Working 

Tooth.  Peculiarities  of  the  Action.  Backs  and  Pinions.  Annular  Wheels. 
Determination  of  Angle  of  Action.  Determination  of  Limiting  Numbers 
of  Teeth.  Wheels  with  Radial  Planes. 

3.  NON-CIRCULAR  SPUR  GEARING.— Construction  of  Teeth  for  Elliptical  and  Lobed 

Wheels. 


Twisted  Spur  Gearing. 

308,  Hooke's  Stepped  Wheels.—  Let  a  pair  of  ordinary  spur  wheels, 
loose  upon  their  shafts,  be  cut  transversely  into  a  number  of  plates. 
Let  these  sections  of  one  of  the  wheels  be  first  rearranged  by  rotating 
them  until,  as  in  Fig.  186,  the  tooth  of  each  overlaps  that  of  the  pre- 
ceding one  by  the  same  amount,  and  then  firmly  keyed  upon  the 
shaft.  In  passing  to  the  new 

1M  .'." 


position,  each  plate  will  drive 
that  section  of  the  other  wheel 
with  which  it  is  in  gear,  inde- 
pendently of  the  others.  Thus 
the  second  series  of  plates  will 
necessarily  be  arranged  in  a 
similar  manner,  and  these 
being  now  also  secured  upon 
their  own  shaft,  we  have  a 
pair  of  the  Stepped  Wheels 
first  introduced  by  Dr.  Hooke. 

By  this  ingenious  device  the  number  of  teeth  is  in  effect  increased 
without  diminishing  their  size.  Thus,  the  figure  shows  a  wheel  built 
up  of  four  plates,  or  thin  wheels  ;  supposing  each  to  have  say  20  teeth, 
the  resulting  action  is  clearly  the  same  as  that  of  a  single  wheel  of 
80,  while  the  fact  that  the  acting  faces  lie  in  different  planes,  enables 


FIG.  186. 


196  HOOKE'S  STEPPED  WHEELS. 

us  to  retain  the  original  pitch.  The  advantages  are  obvious  ;  not 
only  is  the  number  of  contact  points  increased,  but  they  cross  the 
line  of  centres  at  shorter  intervals  ;  and  the  action  is  at  its  best  when 
this  occurs. 

The  teeth  may  have  any  of  the  forms  already  .described  ;  and  the 
extent  to  which  they  overlap  is,  abstractly  speaking,  arbitrary.  But 
clearly  the  best  arrangement  is  to  have  the  edges  of  the  successive 
teeth  divide  the  pitch  arc  AB  into  equal  parts,  as  shown  in  the  figure. 
There  are  in  this  case  four  plates,  and  consequently  the  arc  AC, 
through  which  the  last  plate  is  rotated  from  its  original  position,  is 
equal  to  J  AB.  And  in  general,  letting  n  =  No.  of  Plates,  we  shall 
have 


309.  In  the  practical  employment  of  wheels  thus  constructed  there 
is  a  limit  to  the  reduction  in  the  thickness  of  the  plates  or  steps,  de- 
pending on  the  material  used  and  the  pressure  to  be  transmitted, 
since  if  excessively  thin  they  will   suffer  from  abrasion.     So  long  as 
actual  steps  of  sensible  thickness  are  used,  however,  the  kinematic 
action  differs  in  no  respect  from  that  of  any  other  spur  wheels,  and 
the  lines  of  action  all  lie  in  the  planes  of  rotation. 

If  the  number  of  plates,  then,  be  finite,  it  must  be  comparatively 
small  ;  yet  if  it  be  increased  to  infinity  the  arrangement  again  becomes 
perfectly  practical,  but  the  action  is  modified  in  a  new  and  peculiar 
manner. 

The  steps  now  disappear  entirely,  as  in  Fig.  109  ;  and  the  effect  is 
the  same  as  if  the  original  wheels  had  been  simply  twisted,  as  ex- 
plained in  (175).  In  this  process  the  nature  of  the  acting  surfaces  is 
changed.  «They  were  in  the  first  place  cylindrical,  the  bases  being 
the  involutes  or  epicycloids  forming  the  outlines  of  the  teeth,  and 
the  rectilinear  elements  being  parallel  to  the  axes.  Since  the  twist- 
ing is  uniform,  these  elements  now  become  helices,  all  having  the 
same  pitch,  but  obliquities  differing  according  to  the  distances  from 
the  axes.  And  the  line  of  contact  between  two  engaging  teeth  will 
partake  of  the  helical  form,  though  it  will  not  be  a  true  helix.  For 
it  is  clear  that  the  transverse  sections,  by  successive  planes,  will  be  the 
original  tooth  outlines  in  successive  phases,  and  in  each  section  there 
will  be  a  point  of  tangency,  which  must  lie  in  the  projection  of  the 
path  of  contact  on  a  transverse  plane. 

310.  Now,  in  regard  to  the  common  normal  at  any  point  of  con- 
tact.    Pass  through  the  point  a  transverse  plane,  which  cuts  out  the 


HOOKE'S  SPIRAL  GEARING  NOT  SCREW  GEARING.  197 

tooth  outlines  just  mentioned,  and  draw  also  the  common  tangent  of 
the  two  helices  which  pass  through  the  point ;  these  two  lines  determine 
the  tangent  plane,  and  the  normal  must  be  perpendicular  to  both. 
The  first  lies  in  the  transverse  plane,  but  the  latter  pierces  that  plane 
obliquely.  Consequently,  the  line  of  action  can  in  no  case  lie  in  a 
plane  of  rotation,  btit  will  make  with  it  an  angle  more  or  less  acute. 
In  general,  then,  the  line  of  action  can  be  resolved  into  three  com- 
ponents, viz  : 

1.  The  component  of  rotation,  perpendicular  to  the  plane  of  the 
axes. 

2.  The  component  of  side  pressure,  parallel  to  the  common  perpen- 
dicular of  the  axes. 

3.  The  component  of  end  pressure,  parallel  to  the  axes  themselves. 

When  the  point  of  contact  lies  in  the  plane  of  the  axes,  the  second 
of  these  components  of  course  vanishes  ;  of  which  fact,  as  will  pres- 
ently appear,  advantage  may  be  taken  in  so  forming  the  teeth  that 
there  shall  be  no  sliding  between  them. 

311.  When  the  wheels  are  thus  formed  by  twisting,  instead  of  with 
successive  steps  of  sensible  thickness,  the  combination  is  known  as 
Hooke's  Spiral  Gearing ;  and  is  very  commonly  described  and  classified 
as  a  modification  of  screw  gearing. 

That  this  is  an  error  will  be,  perhaps,  most  clearly  seen  from  the 
considerations,  that  the  direction  of  the  twist  does  not  affect  that  of 
the  rotation,  and  its  amount  does  not  affect  the  velocity  ratio.  Ee- 
garding  the  wheels  as  built  up  of  exceedingly  thin  plates  or  lamina?, 
each  one  of  those  composing  the  driver  turns  the  corresponding  one 
of  the  follower  precisely  as  though  the  thickness  were  sensible,  not 
only  during  the  formative  process  of  twisting  but  after.  On  reach- 
ing the  limit  when  the  plates  become  planes  and  the  elements  of  the 
tooth  surfaces  become  helices,  the  action  is  modified  by  the  deflection 
of  the  common  normals  from  the  planes  of  rotation ;  but  regarding 
the  driver  as  a  screw,  its  endlong  thrust  being  perpendicular  to  those 
planes,  may  have  either  direction  and  any  magnitude  without  affect- 
ing the  direction  or  velocity  of  the  transmitted  motion.  Whereas  in 
screw  gearing  proper,  this  endlong  thrust  either  lies  in  a  plane  of  ro- 
tation or  has  a  component  which  does,  and  this  component  is  the  one 
which  produces  the  rotation  of  the  follower.  In  that  class  of  gearing, 
then,  the  driver  turning  in  a  given  direction,  the  follower  will  turn 
one  way  if  the  driver  be  right  handed,  but  the  other  way  if  it  be  left 
handed  ;  and  the  screw  pitch  of  the  driver  will  obviously  affect  the 
velocity  of  the  imparted  motion. 


198  PRACTICAL  LIMIT  TO   AMOUNT   OF  TWIST. 

312.  From  the  very  nature  of  the  twisting  process,  as  above  ex- 
plained, it  is  evident  that  if  the  screw  pitch  of  one  wheel  of  a  pair  be 
assumed,  that  of  the  other  is  thereby  fixed.     And  it  may  readily  be 
ascertained,  because  if  the  teeth  be  indefinitely  increased  in  number 
and  diminished  in  size,  any  two  which  are  in  contact  will  ultimately 
become  two  tangent  helices  lying  on  the  surface  of  the  original  pitch 
cylinders,  and  must,  therefore,  develope  into  the  same  right  line  on 
the  common  tangent  plane,  as  in  Fig.  32. 

Again,  it  makes  no  difference  in  which  direction  we  twist  the  first 
wheel  ;  but  it  follows  at  once  from  the  preceding  that  if  the  pair  be 
in  outside  gear,  the  helical  elements  of  one  will  be  right  handed, 
those  of  the  other  left  handed.  In  the  case  of  internal  gearing,  on 
the  contrary,  the  helices  of  both  wheels  will  be  either  right  handed 
or  left  handed,  as  the  case  may  be. 

313.  Practical  Choice  of  the  Screw  Pitch.  —  Although,  as  has  been 
stated,  the  amount  to  which  such  wheels  shall  be  twisted  is  abstractly 
optional,  with  the  limitation  mentioned  in  the  preceding  paragraph, 
yet  in  order  to  reduce  the  obliquity  of  action  and  the  consequent  end 
thrust  it  is  best  to  make  it  in  practice  as  small  as  may  be,  consist- 
ently with  securing  the  advantages  due  to  twisting  them  at  all. 

Now  when,  as  thus  far  supposed,  any  of  the  ordinary  forms  of  spur 
teeth  are  used,  the  angle  of  action  is  greater  than  the  pitch,  so  that  in 
any  given  transverse  plane  the  action  of  one  pair  begins  before  that 
of  the  preceding  pair  ends.  Consequently  if  a  wheel  of  any  given 
depth  measured  in  the  direction  of  the  axis  be  tivisted  through  an 
angle  equal  to  the  pitch  of  the  teeth,  all  the  phases  of  the  action  will 
at  all  times  be  simultaneously  presented,  and  there  will  always  be  a 
point  of  contact  in  the  plane  of  the  axes. 

This  then  may  be  taken  as  a  good  practical  limit  ;  and  it  agrees 
with  the  deduction  made  in  (308)  when  the  number  of  steps  becomes 
infinite  ;  —  for  the  expression  there  given, 


gives 

AB 


~  AB  -AC 

whence  if  n  =  oo,  we  have  AC  =  AB. 
314.  Elimination  of  Sliding  Friction. — If  the  teeth  be  so  formed  that 


SLIDING  FRACTION  ELIMINATED. 


199 


FIG.  187. 


in  any  one  plane  the  contact  not  only  begins  but  ends  on  the  line  of 
centres,  continuing  but  for  a  single  instant ;  then  the  point  at  which 
this  contact  occurs,  which  is  of  course  upon  the  surface  of  each  pitch 
cylinder,  will  be  the  only  point  of  tangency  between  the  tooth  surfaces. 
In  the  next  instant  this  driving  point  will  be  found  to  have  shifted 
to  the  consecutive  plane,  and  so  on  continually,  thus  moving  endlong 
in  the  common  element  of  the  pitch 
cylinders.  Under  the  circumstan- 
ces there  is  no  sliding  friction,  since 
the  coincident  points  of  the  acting 
surfaces  are  at  every  instant  moving 
in  the  same  direction  and  at  the 
same  rate. 

In  order  to  comply  with  the  above 
condition,  the  teeth  may  be  formed 
as  shown  in  Fig.  187  or  as  in  Fig. 
188.  In  the  former,  the  outlines 
are  epicycloidal,  but  the  describing 
circles  for  the  faces  are  smaller  than 
those  used  for  tracing  the  correspond- 
ing flanks.  In  the  second  case,  sup- 
pose the  teeth  originally  to  have  been  involutes,  as  shown  in  the  dotted 
lines,  the  flank  proper,  or  part  within  the  pitch  circle,  is  still  of  that 
form,  but  the  face,  or  part  without,  is  of  greater  curvature,  though 
it  is  tangent  to  the  inner  part  at  its  intersection  with  the  pitch  circle. 

In  general,  then,  it  will  be 
easily  seen,  it  is  necessary 
that  when  the  point  of  con- 
tact is  on  the  line  of  cen- 
tres, as  in  either  one  of 
these  figures,  all  the  con- 
tour lines  which  pass  through 
that  point  shall  be  tangent 
to  each  other,  but  the  faces 
must  lie  within  the  curves 
which  in  ordinary  spur  gear- 
ing would  be  conjugate  to 
the  flanks  determined  on. 
A  very  simple  construction 
on  the  epicycloidal  basis  is 
shown  in  Fig.  189,  the  tops  of  the  teeth  being  semicircles  tangent  to 
the  flanks,  which  are  radial,  at  their  extremities. 


FIG.  188. 


200 


END   PRESSURE  NEUTRALIZED. 


The  action  of  plain  spur  wheels  thus  formed  would,  of  course,  be 
correct  only  at  the  instants  here  represented  ;  but  by  the  simple  ex- 
pedient of  twisting  them,  so  that  the  action  on  ceasing  in  one  plane 
is  continued  in  the  next,  these  instants  become  consecutive,  the  ve- 
locity ratio  is  made  constant,  and  the  rotation  is  transmitted  by  pure 
rolling  contact.  On  this  last  account,  as  might  be  expected,  the 
action  of  these  wheels  is  exceedingly  smooth 
and  noiseless,  almost  as  much  so,  if  they  be 
well  made,  as  that  of  belting ;  but  they  are 
better  suited  for  light  work,  because  the  pres- 
sure is  confined  to  a  single  point  instead  of 
being  distributed  along  a  line.  For  heavy  work 
it  is,  therefore,  preferable  to  employ  the  stepped 
wheels,  or  twisted  ones  in  which  the  teeth,  as 
in  Fig.  186,  are  of  the  usual  forms,  although 
the  sliding  friction  still  remains. 

315,  Neutralization  of  End  Pressure. — But  the 
end  pressure  due  to  the  screw-like  action  will 
exist,  even  after  the  forms  of  the  teeth  have 
been  modified  as  above  explained.  What  now 
occurs  during  the  rotation,  as  will  be  evident 
from  a  moment's  study,  is  precisely  equivalent 
to  the  rolling  together  of  two  helices,  one  on  each  pitch  cylinder, 
as  illustrated  in  Fig.  32.  And  this  end  pressure,  which  must  be  re- 
ceived by  a  collar  on  the  bearing,  causes  an  amount  of  friction  which 
it  is  desirable  to  avoid. 

This  disadvantage  may  in  its  turn  be 
obviated  by  the  means  indicated  in  Fig. 
190.     Each  wheel  is  made  in  two  parts  of 
equal  thickness,  which  are  twisted  in  oppo- 
site  directions,  but  to  the  same  extent. 
The  end  thrust  of  one  part,  then,  is  ex- 
actly counterbalanced  by  that  of  the  other  :  FlG-  m 
and  since  the  action  now  involves  pure  rolling  contact  only,  this  may 
be  regarded  as  the  very  perfection  of  geared  wheel- work. 


FIG. 


Pin  Gearing. 

316,  The  term  Epicycloidal  is  in  its  ordinary  use  applied  only  to 
teeth  all  of  which  have  both  faces  and  flanks  whose  describing  curves 
are  circles. 

There  is,  however,  a  form  of  gearing,  in  which  the  teeth  of  one 


PIN   GEARING. 


201 


wheel  of  a  pair  are  true  epicycloids  only  at  a  theoretical  limit,  upon 
reaching  which  those  of  the  other  wheel  become  mathematical 
points.  The  latter  are  in  practice  actually  made  cylindrical  pins 
of  reasonable  diameters,  which  fact  has  given  rise  to  the  name 
Pin  Gearing ;  and  the  former  in  consequence  deviate  considerably 
from  the  epicycloidal  outline  whence  their  working  contours  are 
derived. 

In  Fig.  191,  C  and  D  being  the  centres  of  the  pitch  circles,  let  a 
marking  point  be  fixed  at  A  in  the  circumference  of  the  upper  one. 
Then  while  this  point  goes  to  E,  in  the  progress  of  the  rotation,  it 
will  trace  upon  the  plane  of  the  lower  circle  the  curve  EB  ;  the  arcs 
AE,  AB,  being  equal.  This,  evidently,  is  the  familiar  epicycloid, 


FIG.  191. 


FIG.  192. 


generated  by  rolling  the  upper  pitch  circle  upon  the  lower.  Mean- 
time, since  the  marking  point  does  not  change  its  position  in  the  cir- 
cumference of  the  upper  circle,  it  can  trace  no  curve  at  all  upon  its 
plane. 

Now  let  AF  be  a  curve  similar  to  BE,  and  imagine  a  pin  of  no 
sensible  diameter — a  rigid  material  line — to  be  fixed  at  A  in  the 
upper  wheel.  Then  if  the  lower  one  turn  to  the  right,  it 
will  drive  the  pin  before  it  with  a  constant  velocity  ratio,  the 
action  ending  at  E,  if  the  driving  curve  be  terminated  at  F  as  here 
shown. 

317.  Supposing  A  E  to  be  an  aliquot  part  of  the  circumference,  and 
assuming  it  as  the  pitch  arc,  we  have  only  to  set  the  pins  at  equal 


202  FORMATION   OF   THE  TEETH. 

distances  in  the  upper  circle,  and  after  dividing  the  lower  one,  to 
draw  through  the  points  of  division  the  reversed  curves  as  shown  : 
this  done,  the  elementary  wheels  are  complete  and  capable  of  working 
in  either  direction.  From  these  the  practical  ones  are  derived  as  in 
Fig.  192  ;  the  pins  being  made  of  sensible  magnitude,  the  outlines 
of  the  teeth  upon  the  other  wheel  are  curves  parallel  to  the  original 
epicycloids.  The  diameter  of  the  pins  is  usually  about  equal  to  the 
thickness  of  the  teeth  measured  on  the  pitch  circle,  the  radius  being, 
therefore,  one  fourth  of  the  pitch  arc  ;  this,  however,  is  not  impera- 
tive, and  the  pins  are  sometimes  made  considerably  smaller.  When 
the  radius  has  been  selected,  a  number  of  arcs  are  described  with  it, 
having  their  centres  upon  the  epicycloid,  and  the  envelope  of  these 
arcs  is  the  required  contour  of  the  working  tooth. 

The  pins  are  ordinarily  supported  at  each  end,  two  discs  beins: 
fixed  upon  the  shaft  for  the  purpose,  as  in  Fig.  Ill  :  thus  making 
what  from  its  form  is  called  a  lantern  wheel  or  pinion. 

318.  Peculiarities  of  the  Action. — The  most  striking  feature  of  wheel- 
work  of  this  kind  is,  that  the  action  is  almost  wholly  confined  to  one 
side  of  the  line  of  centres. 

In  the  elementary  form,  this  is  a  direct  and  obvious  consequence 
of  the  manner  in  which  the  tooth  is  generated.  Thus,  in  Fig.  191, 
if  the  curve  AF  drive,  the  action  cannot  begin  until  its  root  reaches 
the  point  A  on  CD,  and  is  entirely  receding ;  if  on  the  other  hand 
this  curve  be  driven  by  the  pin,  the  action  will  terminate  at  the  same 
point  A,  and  will  be  entirely  approaching. 

Consequently,  pin-gearing  is  not  well  adapted  for  use  in  combina- 
tions which  require  the  same  wheel  both  to  drive  and  to  follow.  But 
when  that  is  not  required,  this  peculiarity  of  the  action  is  greatly  in 
its  favor ;  the  pins  of  course  being  always  given  to  the  follower  and 
the  teeth  to  the  driver.  And  previously  to  the  introduction  of  cut 
gearing,  it  was  very  extensively  employed,  even  for  heavy  work  ;  the 
facility  of  forming  the  pins,  or  staves,  of  the  lantern  pinions,  in  the 
lathe,  especially  adapting  it  for  the  construction  of  wooden  wheels. 
At  present  its  use  is  almost  exclusively  confined  to  clock-work  and 
similar  light  mechanism,  the  pins  being  made  of  steel  wire  cut  to  the 
proper  lengths. 

319.  It  has  heretofore  been  taught,  in  all  the  treatises  upon  this, 
subject  which  have  come  to  our  notice,  that  when  the  pins  are  made 
of  sensible  diameter,  as  in  working  wheels  they  always  must  be,  the 
above-mentioned  peculiarity  is  modified  and  a  certain  amount  of  ap- 
proaching action  introduced  in  every  case. 

The  fact  was  too  patent  to  escape  notice,  that  the  expansion  of  the 


BACKS 


PIN   GEAKING. 


203 


pin  from  the  theoretical  point  into  the  practical  circle,  had  the  result 
of  shortening  the  driver's  tooth  and  reducing  the  amount  of  receding 
action.  But  while  this  palpable  effect  of  the  new  condition  upon  the 
time  when  the  action  shall  end  was  duly  recognized,  no  one  seems 
ever  to  have  inquired  whether  the  same  condition  might  not  also  affect 
the  time  when  the  action  shall  begin ;  and  it  appears  to  have  been 
taken  for  granted  that  it  would  not. 

Thus  in  Fig.  192,  the  pin  E  is  just  quitting  contact  with  the  tooth 
which  drives  it ;  and  the  theory  as  hitherto  laid  down  is  based  upon 
the  assumption,  that  correct  driving  contact  with  the  next  pin  A  is 
just  beginning  when  it  occupies  the  position  here  shown  ;  thus  giving 
an  arc  of  approach  about  equal  to  the  radius  of  the  pin,  and  making 
this  as  it  stands  an  exact  limiting  case. 

It  will  presently  be  shown  that  this  assumption  is  erroneous,  and 
that  the  next  tooth  is  not,  in  general,  tangent  to  the  pin  A  when  in 


FIG.  193. 


FIG.  194. 


its  present  position.  The  error  itself  is  physically  small,  which  may 
account  for  its  having  remained  so  long  undetected  ;  but  its  effect  is 
of  very  perceptible  magnitude  ;  not  only  changing  very  greatly  the 
amount  of  approaching  action,  but  in  many  cases  making  that  action 
absolutely  negative,  the  first  driving  contact  between  each  tooth  and 
its  pin  not  occurring  until  after  the  latter  has  bodily  passed  the  line 
of  centres. 

320.  Pin- Wheel  and  Rack. — Since  the  pins  are  always  given  to  the 
follower,  the  construction  of  a  rack  will  present  two  cases.  If  it  is  to 
be  the  driver,  as  in  Fig.  193,  the  elementary  tooth  is  bounded  by 
cycloids,  generated  by  the  pitch  circle  of  the  wheel  ;  from  which  the 
outlines  of  the  working  tooth  are  derived  as  above  explained. 

Pin-Rack  and  Wheel,— In  Fig.  194,  the  wheel  is  to  drive,  and  the 
tooth-outline  is  the  involute  of  its  own  pitch  circle,  the  generatrix 
being  the  pitch  line  of  the  rack.  It  is  usually  stated  that  it  is  unnec- 


204 


ANNULAR  WHEELS  IK   PIN   GEAKING. 


essary  to  construct  the  derived  curve,  since  "  this  process  would  merely 
reproduce  the  same  involute  in  a  different  position."  *  Although  this 

is  the  truth,  it  is  not  the 
whole  truth  ;  and  the  part 
which  is  lacking,  will  be 
found  to  vitiate  the  results 
deduced  from  the  part  which 
is  given. 

321.  Inside  Pin  Gearing.— 
Here  also  we  have  t\vo  dis- 
tinct cases,  since  the  annular 
wheel  may  be  required  either 
to  drive  or  to  follow.  In 
Fig.  195,  the  pinion  drives, 
and  the  pins  are  given  to 
the  wheel.  The  outer  pitch 
circle,  then,  by  rolling  upon 

the  inner,  generates  the  internal  epicycloid  which  forms  the  element- 
ary tooth  of  the  pinion.  In  Fig.  196,  the  wheel  drives,  and  its  ele- 
mentary tooth  is  the  hypocycloid  traced  by  rolling  the  less  circle 
within  the  greater  ;  in  both  cases  the  curves  for  the  working  teeth  are 
derived  in  the  usual  manner.  If  the  annular  driver  be  twice  as  large 


195. 


FIG.  196.  FIG.  107. 

as  the  pinion,  the  hypocycloid  becomes  a  right  line,  and  the  process 
of  derivation  gives  simply  another  right  line  parallel  to  it.  A  very 
practical  construction  in  this  case  is  shown  in  Fig.  197  ;  the  pinion  has 


*  Willis.     (Principles  of  Mechanism,  p.  96.) 


WHAT   CONDITIONS   AEE   PRACTICABLE. 


205 


but  two  pins,  which  turn  in  blocks  sliding  in  the  two  slots  which,  cross- 
ing each  other  at  right  angles,  constitute  the  disguised  annular  wheel. 

322.  Practicability  of  Assumed  Conditions. — It  is  hardly  necessary  to 
state  that  there  are  definite  relations  between  the  pitch  arc  and  arc 
of  action,  the  diameter  of  the  pin  and  the  height  of  the  tooth,  such 
that  all  these  cannot  be  assumed  with  any  certainty  that  the  result 
will  be  a  practicable  arrangement.  The  preceding  figures  give  merely 
an  idea  of  the  general  principles  and  the  general  appearance  of  the 
various  modifications  ;  and  we  have  now  to  consider  more  in  detail 
the  processes  of  construction. 

First,  in  regard  to  the  elementary  or  ideal  form,  in  which  the  pin 
is  a  point  and  the  tooth- 
curve  a  true  epicycloid. 
Eeferring  to  Fig.  191,  it 
appears  that  if  we  as- 
sumed the- pitch  arc  AS, 
the  greatest  possible 
height  of  the  tooth  is  de- 
termined by  the  intersec- 
tion of  the  front  and  back 
at  G  ;  and  if  this  height 
be  adopted,  the  action, 
beginning  at  A,  will  ter- 
minate at  H,  the  point 
on  the  upper  pitch  cir- 
cumference through 
which  G  must  pass  in  its 
rotation  about  D.  Now 
this  point  G  lies  on  the 
radial  line  which  bisects 
AB,  and  will,  therefore, 
fall  farther  to  the  right, 
the  larger  the  pitch  arc. 
Should  it  chance  to  coin- 
cide with  7f,  or  in  other 
words  should  the  point  FIG.  198. 

of  the  driver's  tooth  fall  upon  the  pitch  circle  of  the  follower,  the 
proposed  case  is  exactly  a  limiting  one.  From  an  examination  of  the 
succeeding  figures  it  will  be  seen  that  this  is  equally  true  when  the 
driver  is  a  rack  or  an  annular  wheel ;  and  also  that  in  order  to  render 
the  case  practically  feasible,  the  point  G  should  fall  within  the  pitch 
circle  of  the  follower,  except  when  the  latter  is  annular,  in  which 


206 


LIMITING   DIAMETER   OF   PITS. 


case  it  should  fall  without.  This  condition  being  satisfied,  the  tooth 
may  be  made  pointed,  or  it  may  be  topped  off,  at  pleasure ;  and 
finally,  the  arc  of  action  is  determined,  as  in  previous  constructions,  by 
the  intersection  of  the  path  of  the  highest  point  of  the  tooth,  with  the 
path  of  contact— that  is  to  say  with  the  pitch  circle  of  the  pin- wheel. 
323.  Second,  when  the  pin  is  of  sensible  diameter.  We  give  in 

Fig.  198,  Prof.  Willis's 
mode  of  determining  the 
greatest  diameter  which 
can  be  used  under  any 
assumed  conditions.  C and 
D  being  the  centres  of  the 
follower  and  driver  re- 
spectively, suppose  the 
pitch  arc,  AB  or  AE,  to 
be  assigned.  Bisect  AB 
by  the  indefinite  radial 
line  Dx,  and  draw  the 
chord  AE,  cutting  Dx  in 
/ ;  then  El  is  the  greatest 
radius  for  the  pin,  and 
any  less  one  may  be  used, 
as  in  Fig.  192.  Since  / 
is  unquestionably  the  last 
point  of  contact  with  the 
pin  E,  this  construction 
would  be  correct,  provided 
that  the  next  tooth  KM 
were  now  in  driving  con- 
tact with  the  pin  A.  Prof. 
Willis  distinctly  states  that 
it  is  ;  *  and  bases  this  as- 
sertion upon  the  previous  statement  f  that  "  by  the  mode  of  its  de- 
scription the  circle  (of  the  pin)  must  touch  the  curve  (of  the  tooth) 
when  its  centre  is  in  any  point  of  the  epicycloid."  The  conclusion, 
therefore,  is  very  plausible ;  but  it  is  not  correct,  because  the  curve 
derived  from  the  epicycloid  AR  (as  was  pointed  out  in  Fig.  178), 
consists  of  two  branches,  one  of  which,  and  a  part  of  the  other  also,  is 
effaced  in  the  very  process  of  formation.  The  normal  to  the  epicy- 


Fio.  199. 


*  Principles  of  Mechanism,  p.  99. 
f         "         "          "  p.  94. 


LIMITING   DIAMETER   OF   PIN. 


UNIVERSITY 

207 


cloid  at  its  root,  is  AO  perpendicular  to  CD  ;  the  parallel  CUIM 
at  first  descends,  having  a  vertical  tangent  at  0,  forms  a  cusp  at  a 
point  within  the  pin  whose  centre  is  A9  and  then  rising,  cuts  its  cir- 
cumference at  some  point  M,  which  is  evidently  the  lowest  practicable 
point  in  the  outline  of  the  working  tooth.  Draw  through  M  a  nor- 
mal, cutting  the  original  epicycloid  at  N,  and  through  that  point  de- 
scribe an  arc  about  D,  cutting  the  locus  of  original  contact  in  P.  It 
will  now  be  seen  that  the  point  M  cannot  come  into  correct  driving 
contact,  until  N  reaches  the  position  P\  and  the  arc  AP  is  the 
measure  of  the  error  in  Prof.  Willis's  construction  ;  that  is  to  say,  it 
is  the  amount  by  which  the  arc  of  approach  as  determined,  or  rather 
assumed  by  him,  must  be  changed. 

324.  Absolute  Determinations  Impossible, — It  appears  then  that  the 
limiting  diameter  of  the  pin  as  found  by  the  above  construction  is 
too  great.  By  reducing  it,  the  tooth  IL  may  be  made  enough  longer 
to  continue  in  action  until  the  point  M  comes  into  driving  contact. 
But  the  precise  amount  of  reduction  which  will  cause  one  tooth  to 
quit  contact  at  the  instant  when  the  next  one  begins  to  drive,  it  is 
impossible  to  determine  ;  because  the  positions  of  both  the  points  M 
and  /,  which  fix  the  times  of  the  beginning  and  the  ending  of  the 
action,  depend  upon  the  unknown  magnitude,  the  radius  of  the  pin. 
We  are,  therefore,  compelled  to  adopt  the  tentative  process  shown  in 
Fig.  199.  Having  found  the  point  /,  cio  in  the  preceding  diagram, 
assume  a  radius  for  the  pin,  less  than  El,  and  continue  the  derived 
curve  to  cut  Dx  in  /,  which  will  be  the  point  of  the  tooth.  Through 
J  'draw  a  normal  to  the  epicycloid,  cutting  it  in  8 ;  through  8  de- 
scribe an  arc  about  D,  which  will  cut  the  upper  pitch  circle  in  T,  the 
position  of  the  pin  at  the  end  of  the  action.  Drawing  the  working 
outline  of  the  next  tooth,  we  determine,  as  in  Fig.  198,  the  points  M, 
N,  and  finally  P,  the  position  of  the  pin  when  the  action  begins.  Now, 
if  the  arc  PT  prove  to  be  precisely  equal  to  AE,  we  have  an  exact  lim- 
iting case,  and  the  assumed  radius  of  the  pin  is  a  maximum  :  if  PT 
be  less  than  AE,  the  radius  is  too  great,  and  must  be  reduced — but  if 
the  contrary,  the  case  is  a  practicable  one. 

In  the  latter  event  the  tooth  may  be  topped  off  ;  and  it  need  hardly 
be  added  that  the  above  process  enables  us  to  determine  whether  the 
case  be  feasible  or  not,  if  both  the  diameter  of  the  pin  and  height  of 
the  tooth  are  assigned  ;  the  normal  JS  being  drawn  from  the  highest 
point  of  the  given  curve.* 

*  Note.  For  details  of  the  graphic  processes  of  drawing  these  curves  and  the 
normals,  the  reader  is  referred  to  the  Appendix. 


208 


PIN   GEARING. 

Limiting  Numbers  of  Teeth  and  Pins. 


325.  When  the  Pin  is  a  Mathematical  Point,  the  determination  of 
the  limiting  numbers  of  teeth  and  pins  is  easily  effected  by  methods 
precisely  like  those  used  in  the  case  of  the  common  epicycloidal  teeth. 
Suppose  the  number  of  teeth  to  be  assigned  for  the  driver  whose  centre 
is  b  in  Fig.  200,  and  let  AFbe  the  pitch  arc.  The  point,  0,  of  the 

tooth  must  lie  on  the  pro- 
longation of  DL  bisecting 
AF,  in  such  a  position 
that  an  arc  OA  of  a  circle 
whose  centre  lies  upon  DA 
produced,  shall  be  equal  to 
the  arc  AF.  On  the  tan- 
gent at  A,  set  off  AG  = 
arc  AF,  and  A M  =  J  A  G: 
with  centre  M  and  radius 
MG  describe  an  arc  cutting 
DL  produced,  in  0 ;  draw 
OA,  and  bisect  it  by  a  per- 
pendicular NC,  which  cuts 
DA  or  its  prolongation  in 
C,  the  required  centre  of 
the  follower. 

In  this  diagram,  the 
wheels  evidently  work  in  outside  gear,  and  A  (7 is  a  minimum.  Should 
0  fall  upon  the  tangent  A  G,  OA  will  coincide  with  that  line,  and 
the  follower  becomes  a  rack  :  if  0  falls  above  A  G,  the  centres  C  and 
D  will  lie  on  the  same  side  of  the  point  of  contact,  the  follower  will 
be  annular,  and  the  radius  A  C  thus  found  will  be  a  maximum. 

If,  on  the  other  hand,  the  number  of  pins  be  assigned  for  the  fol- 
lower ;  then  the  position  of  0  is  known,  and  also  the  arc  A  0,  whence 
AG  and  AM  are  also  determined.  It  is  evident  that  OD  when  found 
will  bisect  the  chord  AF,  making  OA  =  OF.  Therefore,  describe 
about  J-fan  arc  with  radius  MG,  and  about  0  another  arc  with  radius 
OA  ;  these  arcs  will  intersect  in  F,  and  a  perpendicular  to  AF 
through  0  will  cut  CA  or  its  prolongation  in  D,  the  required  centre 
of  the  driver,  whose  radius  will  be  a  minimum  when,  as  in  this  figure, 
A  lies  between  C  and  D. 

If  the  point  F  coincides  with  G,  we  have  the  limiting  case  when 
the  driver  becomes  a  rack  :  should  F  fall  below  A  G,  the  driver  becomes 
in  its  turn  annular,  and  its  radius  as  thus  determined  is  a  maximum. 


FIG.  200. 


KADIAL   FLAKES — LIMITING   EATIO    OF   PITCH   CIRCLES. 


209 


326.  Wheels  with  Kadial  Planes. — Since  the  face  OF  of  the  driver's 
tooth  in  Fig.  200  is  an  epicycloid,  it  would  correctly  drive  a  flank 
traced  by  the  same  describing  circle.     And  this  flank  would  be  radial, 
were  we  to  use  a  pitch  circle  whose  radius  AE  is  twice  AC.     Hence 
this  driver  will  work  not  only  with  the  pin-wheel,  but  with  a  wheel 
of  twice  the  diameter  of  the  latter,  furnished  with  twice  as  many 
radial  planes  as  there  are  pins :  the  combination  presenting  the  ap- 
pearance shown  in  Fig.  201. 

In  other  words,  the  minimum 
number  of  radial  planes  which 
can  be  driven  by  such  a  wheel 
is  equal  to  twice  the  minimum 
number  of  pins.  When  stated 
in  this  way,  it  would  appear 
that  there  should  always  be  an 
even  number  of  these  planes. 
But  it  is  to  be  observed,  that 
in  computing  the  least  number 
of  pins,  we  must  take  the  next 
higher  integer  in  the  event  of 
a  fractional  result ;  we  may, 
however,  at  once  double  that 
result  in  order  to  ascertain  the 
least  number  of  radial  planes, 
and  the  next  higher  integer  is 
not  then  necessarily  an  even  number ;  but  on  the  contrary,  is  an 
odd  number  in  the  majority  of  cases,  as  will  be  seen  by  reference  to 
the  following  tables. 

327.  Limiting  Ratio  of  Pitch  Circles  in  Inside  Gear. — In  the  use  of 
the  ordinary  epicycloidal  teeth,  or  of  those  of  involute  form,  the  di- 
ameters of  wheels  in  inside  gear  may  be  made  very  nearly  equal  if  the 
teeth  are  short  enough,  as  in  Fig.  129  :  and  such  combinations  are 
frequently  met  with   in  differential  trains,  the  annular  wheel  often 
having  only  one  tooth  more  than  the  pinion.     And  at  first  glance  it 
would  appear  perfectly  feasible  to  adopt  such  proportions  in  pin  gear- 
ing, whether  the  annular  wheel  drive  or  follow. 

But  in  the  former  case,  the  teeth  being  hypocycloids  of  which  the 
inner  pitch  circle  is  the  generatrix,  will  become  radial  when  the  outer 
one  is  of  twice  its  diameter  ;  and  if  this  limit  be  passed,  making  the 
pinion  more  than  half  as  large  as  the  driver,  the  teeth  of  the  latter 
will  become  concave :  this,  although  geometrically  satisfying  the  con- 
ditions, is  manifestly  impracticable. 
14 


FIG.  201. 


210 


LIMITING   NUMBERS   OF   TEETH. 


If,  however,  the  internal  wheel  drive,  no  such  difficulty  is  met  with, 
and  it  may  have  nearly  as  many  teeth  as  there  are  pins  in  the  annular 
follower,  if  they  be  duly  shortened  as  the  number  is  increased. 

328.  When  the  Pin  is  of  Sensible  Diameter,  the  action,  as  has  been 
shown,  begins  at  a  different  time  ;  and  in  consequence  of  the  peculiar 
nature  of  the  derived  curve  forming  the  working  outline,  it  becomes 
in  general  impossible  to  determine  the  limiting  numbers  with  precis- 
ion.    For  if  the  diameter  of  the  pin  be  assumed,  then  because  one 
pitch  circle  is  the  generatrix  and  the  other  is  the  directrix  of  the  ele- 
mentary tooth,  the  diameters  of  both  must  be  known  before  the  point 
Mot  Figs.  198  and  199,  which  fixes  the  time  when  the  action  shall 
begin,  can  be  found  graphically  or  otherwise. 

In  one  case,  however,  this  point  can  be  located  with  exactness,  as 
shown  in  Fig.  202 ;  DD,  CC,  being  arcs  of  the  pitch  circles  of  the 

driver  and  the  follower  re- 
spectively, and  having  the 
same  radius.  Let  AO  be 
the  radius  of  the  pin,  which 
is  here  enormously  exag- 
gerated ;  CC  cuts  its  cir- 
cumference at  N  and  L, 
and  DD  cuts  it  at  M  and  S, 
making  the  arcs  AN,  AM, 
equal  to  each  other.  Rolling 
CC  upon  DD,  the  point  A 
rises  in  the  cardioidal  curve 
AB,  and  when  N  reaches 
M,  the  chord  AN  will 
have  the  position  MB,  nor- 
mal to  this  curve  and  also 
to  the  derived  one. 

The  latter  is  composed  of  the  two  branches  OP,  PE ;  the  second 
one,  therefore,  cuts  the  base  circle  and  the  circumference  of  the  pin, 
whatever  the  radius  of  the  latter,  at  the  same  point,  M.  Since  MA  = 
NA,  the  points  M  and  j^will  come  together  at  A  as  the  rotation  in- 
dicated by  the  arrow  progresses  :  and  moreover,  M  will  then  be  in 
correct  driving  contact,  the  normal  MB  at  that  instant  coinciding 
with  the  chord  AL. 

329,  We  have  thus  ascertained,  that  when  the  driver  and  the  fol- 
lower are  of  equal  diameters,  there  is  no  arc  of  approach,  the  action 
beginning  on  the  line  of  centres,  without  regard   to  the  size  of  the 
pin.     "We  may  now  assume  a  diameter  for  the  latter,  and  by  a  slight 


FIG.  202. 


DETERMINATION   OF   LIMITING   NUMBERS. 


211 


modification  of  preceding  processes  we  can  determine  the  least  num- 
ber of  pins  and  teeth  that  can  be  used. 

Let  us  suppose  the  pin  to  be  of  such  size  that  the  arc  NL  of  Fig. 
202  shall  be  one  half  the  pitch  ;  then  in  Fig  203  we  shall  have  AO  = 
li  pitch,  AL  =  pitch,  and  AF  —  1J  pitch  ;  and  the  point  P,  of  the 
working  tooth  must  at  the  end  of  the  action  lie  at  the  intersection  of 
the  chord  OA  with  the  radial  line  DP  which  bisects  AF. 

Now,  were  it  true  that  the  arc  of  approach  is  nil  in  all  cases,  the 
limiting  numbers  could  be  de- 
termined. The  graphic  pro- 
cess is  as  follows.  First,  if  the 
number  of  teeth  be  assigned, 
we  make  the  angle  ADP  =  f 
pitch,  set  off  on  the  tangent  at 
A,  a  distance  AG  equal  to  the 
pitch  arc  AL,  make  AM  =  J 
A  G,  then  with  centre  M  and 
radius  MG  describe  an  arc  cut- 
ting the  radial  line  DP  in  P. 
The  angle  GAP  thus  deter- 
mined is  evidently  equal  to 
ACN,  and,  therefore,  equal  to 
-f  of  the  pitch  angle  of  the  fol- 
lower, whence  the  required 
Jninimum  number  of  pins  is 
easily  found.  Second,  if  the 
number  of  pins  be  assigned, 
then  PA  is  known,  and  PF 
must  be  equal  to  it;  also 'the 
arc  A  F  when  found  must  be  1|  times  the  pitch  arc  ;  this  is  of  a 
known  length,  and  having  set  off  AH  equal  to  it  on  the  tangent,  we 
can  find  F  as  in  previous  cases,  thus  determining  the  required  least 
number  of  teeth,  which  could  drive  the  given  follower  on  the  suppo- 
sition that  there  is  no  approaching  action. 

330.  We  must  now  proceed  tentatively,  for  this  supposition  is  true 
only  in  the  one  case  in  which  there  are  just  as  many  teeth  as  there  are 
pins.  If  then  we  assume  various  numbers  of  either,  the  minimum 
values  as  thus  determined  will  in  general  be  incorrect ;  but  the  errors 
will  diminish  as  the  number  sought  approaches  equality  with  the 
number  assumed,  and  when  that  equality  is  reached  the  result  will 
be  exact. 

In  this  way  we  find,  by  computation, 


FIG.  203. 


212 


PIN"   GEARING. 


Fid.  204. 


Least  No.  of  Pins  for  12  Teeth  —  11.62. 
Least  No.  of  Teeth  for  12  Pins  =  11.77. 

Twelve  teeth,  then,  can  be  used  to  drive  twelve  pins,  but  no  less 
numbers  will  answer  for  equal  wheels  when  the  pin  is  of  the  size  above 
assumed,  which  is  that  most  commonly  adopted  in  practice.  If  the 
pin  be  made  smaller  these  numbers  may  be  reduced,  the  limit  being 
six  for  each  wheel  when  the  pin  becomes  a  point. 

331,  The  above  process,  with  slight  and  obvious  modifications  in  the 

diagram,  Fig.  203, 
would  be  applicable 
in  every  case,  were  the 
amount  of  approach- 
ing action  known  for 
every  given  ratio  of 
the  pitch  diameters. 
But  it  is  not  known; 
and  further  progress 
must  be  made  in  the 
face  of  additional 
obstacles  due  to  the  perverse  nature  of  the  derived  curve.  Consider- 
ing these  now  in  reference  to  outside  gear,  we  find  that  when  the 
pitch  circles  are  unequal,  the  point  M of  Fig.  202,  in  which  this  curve 
cuts  the  circumference  of  the  pin,  no  longer  falls  upon  the  base  circle 
of  the  epicycloid.  If  the  driver  be  the  larger,  the  second  branch  PE 
cuts  the  pin  first,  as  in  Fig.  204,  which  represents  the  limiting  case 
of  a  driving  rack,  subsequently  intersecting  the  base  line  at  some 
point  JR.  The  latter  will  come  into  driving  contact  when  it  has  ad- 
vanced to  A,  but  before  that  occurs  the  normal  MB  will  have  reached 
the  position  M'AB',  thus  bringing  the  point  Jf  into  action,  and  giv- 
ing a  greater  or  less  arc  of  approach.  If,  on  the  other  hand,  the  fol- 
lower be  the  larger,  the  working  branch  of  the  curve  will  cut  the  base 
circle  before  it  does  the  circumference  of  the  pin.  The  former  inter- 
section is  the  one  which  would  come  into  action  on  the  line  of  centres, 
but  since  it  is  effaced  in  the  process  of  forming  the  actual  tooth,  it 
does  not  come  into  action  at  all,  and  the  latter  intersection  cannot, 
evidently,  begin  to  drive  until  after  passing  that  line,  thus  introduc- 
ing the  phenomenon  of  negative  approach. 

The  limiting  case  in  outside  gear  is  that  of  the  driven  pin-rack,  Fig. 
205  ;  it  is  very  true  that  the  process  of  constructing  the  derived  curve 
"  reproduces  the  same  involute  in  a  different  position  "  (PE},  but  it 
also  introduces  a  part  (PO)  of  the  other  branch ;  to  be  sure,  this  is 


METHOD   OF   COMPUTING   TABLES. 


213 


effaced,  but  so  too  is  a  portion  (PM)  of  what  would  be  the  acting 
tooth  ;  and  M  cannot  drive  until  it  reaches  the  position  B'  in  the 
path  of  contact  CC,  the  negative  approach  being,  therefore,  equal  to 
AB'. 

332.  Now,  the  exact  mathematical  determination  of  this  lowest 
point  M9  even  when  the 
diameters  of  both  pitch 
circles  and  the  pin  are 
given,  is  a  matter  of  ex- 
treme complexity,  to  say 
the  least  of  it.  And  were  Q_ 
it  never  so  simple,  that 
fact  would  be  of  service 
only  in  determining  the 
limiting  numbers  by  a 
series  of  approximations, 
since  when  one  pitch  circle 
is  given  the  other  is  to  be 
found.  FIG- 205- 

Still,  these  numbers  may  be  ascertained  with  a  degree  of  precision 
sufficient  for  all  practical  purposes,  we  think,  by  the  method  which 
we  have  adopted  in  computing  the  subjoined  tables.  This  is  based 
upon  the  assumption  that  the  amount  of  approaching  action,  positive 
or  negative,  can  be  determined  as  accurately  as  necessary  by  grapliic 
processes.  Which  will  appear  the  more  probable,  when  it  is  taken 
into  account  that  in  the  actual  working  tooth,  the  obtuse  intersection 
M  will  be  more  or  less  rounded  off,  and  not  sharply  defined  ;  and  the 
effect  of  this  will  be  to  diminish  the  amount  of  positive,  and  to  in- 
crease that  of  negative,  approach. 

Accordingly,  it  appearing  from  preliminary  trials  that  about  six 
teeth  will  be  the  least  that  can  drive  a  pin-rack,  an  accurate  diagram 
upon  a  large  scale  (the  diameter  of  the  pinion  being  six  feet)  gave  for 
a  pin  of  the  proportions  named  in  (329),  a  negative  approach  equal 

to  ~-TT  •  Then,  assuming  T^  the  pitch  as  the  actual  amount,  we 
have  the  equation 


or 


tan  (f-  pitch)  =  arc  (1  +  T1^)  pitch, 
tan  10z°  =  arc  13z°  ; 


which  being  solved  by  the  tentative  process  previously  described,  gives 
the  limiting  number,  6.44,  for  the  pinion  driving  a  rack. 


214 


GEAKING. 


Seven  teeth,  then,  will  drive  in  outside  gear  ;  the  negative  approach 
would  theoretically  be  a  little  less  —  but  in  view  of  the  consideration 
above  mentioned  we  have  not  reduced  the  amount,  but  have  allowed 
-j*g-  the  pitch  in  this  case  also.  We  have  previously  found  that  12 
teeth  will  drive  12  pins,  the  approach  being  exactly  zero  :  what  re- 
mains,, then,  is  to  take  for  the  drivers  between  7  and  12,  a  negative 
approach  diminishing  as  the  numbers  of  teeth  increase,  and  to  test 
the  results  of  computation  by  graphic  construction. 

333.  Similarly,  we  find  in  the  case  of  a  rack  driving  a  pin-wheel,  a 
positive  approach  of  about  T\-  pitch;    which  gives  4.64  as  the  least 
number  of  pins  that  can  be  used.     As  the  amount  of  approaching 
action  here  named  was  determined  graphically  for  a  wheel  of  six  pins, 
it  is  probably  less  than  it  would  actually  be  at  the  exact  limit,  and 
we  have  allowed  the  same  amount  in  the  case  of  a  wheel  of  5  pins, 
which,  on  this  hypothesis,  can  be  driven  by  one  of  110  teeth.     For 
the  pin-  wheels  between  12  and  5,  then,  we  have  a  positive  approach, 
gradually  increasing  from  zero  to  y\  of  the  pitch. 

We  repeat,  that  no  pretension  to  theoretical  exactness  is  here  made, 
the  object  being  to  furnish  safe  rules  for  practical  guidance  in  using 
the  customary  proportions.  We  have,  therefore,  in  apportioning  the 
positive  and  negative  approach,  made  the  former  rather  less  and  the 

latter  rather  greater  than  the 
precise  amount  would  proba- 
bly be.  It  may  be  observed, 
that  the  construction  of  the 
nomodont  for  the  whole  range 
from  the  driving  to  the  driven 
rack,  affords  a  tolerably  reli- 
able check  upon  the  grada- 
tion of  the  approaching  action, 
since  any  serious  error  would 
be  manifested  by  the  conse- 
quent irregularity  of  the 
curve.  Besides,  the  graphic 
constructions  have  been  made 
with  considerable  care  and 
upon  a  large  scale  ;  so  that  upon  the  whole  we  think  it  safe  to  say 
that  if  the  radius  of  the  pin  be  in  each  case  made  equal  to  half  the 
chord  of  the  pitch  arc  upon  the  smaller  circle,  the  numbers  here 
given  will  work  ;  and  there  is  a  strong  probability  that  these  limits 
cannot  be  practically  transcended. 

334.  Next,  in  regard  to  inside  gear.      When  the  annular  wheel 


E 


FlG-  206- 


DETERMINATION   OF   LIMITING    NUMBERS. 


215 


drives,  the  positive  approach  is  still  greater  than  with  the  driving 
rack,  as  will  be  seen  by  comparing  Fig.  204  with  Fig.  206  ;  the  point 
R,  which  will  come  into  driving  contact  with  N,  at  A  upon  the  line 
of  centres,  being  in  the  latter  case  much  farther  from  M,  the  lowest 
practicable  point  of  the  working  tooth. 

This  increase  in  the  approaching  action  reaches  its  limit  when  the 
diameter  of  the  larger  pitch  circle  is  twice  that  of  the  smaller ;  the 
hypocycloid  then  becomes  a  right  line,  the  derived  curve  becomes 
another  one  parallel  to  it,  and  whatever  the  position  or  the  size  of  the 
pin,  the  common  perpendicular  to  these  two  lines  through  its  centre 
always  passes  through  A.  In  the  case  represented  in  Fig.  197,  then, 
the  approaching  action  continues  during  half  the  revolution  of  the 


FIG.  207. 

follower,  and  the  receding  action  during  the  other  half,  each  being, 
therefore,  equal  to  the  whole  pitch.  A  greater  number  of  pins  and 
slots  may  be  used,  but  the  sliding  blocks  must  then  be  dispensed  with, 
and  the  -angle  of  action  will  be  reduced,  its  magnitude  depending 
upon  the  distance  from  the  driver's  centre  at  which  the  slots  terminate 
or  intersect  each  other  ;  that  is,  in  effect,  upon  the  length  of  the  teeth. 
335.  On  the  other  hand,  when  the  pinion  drives,  the  negative  ap- 
proach becomes  greater  than  in  the  case  of  a  pin-rack.  And  here  too 
an  exact  determination  of  its  amount  can  be  made  when  the  diameter 
of  the  outer  circle  is  twice  that  of  the  inner,  as  in  Fig.  207 ;  for  the 
original  tooth-outline  AB  is  in  this  case  also  a  cardioid,  and  the  lowest 
point  M  of  the  working  tooth  falls  upon  the  pitch  circle,  DDy  of  the 
pinion.  But  the  driver  must  now  make  two  revolutions  in  order  to 
have  the  complete  cardioid  traced  upon  its  plane  by  a  marking  point 
in  the  circumference  of  the  follower,  whereas  when  two  equal  wheels 
work  in  outside  gear  but  one  revolution  is  necessary  :  and  the  same 


216 


PIN"   GEARING. 


proportion  holds  true  for  the  generation  of  any  given  portion  of  the 
curve.  Therefore,  instead  of  coming  into  driving  contact  at  A,  as  in 
Fig.  202,  the  point  M  must  advance  twice  as  far  before  its  action 
begins  ;  the  point  B  comes  into  the  original  locus  of  contact  at  B', 
the  normal  MB  at  that  instant  taking  the  position  SB' :  thus  fixing 
the  arc  of  negative  approach  AS,  which  is  readily  computed  when 
the  size  of  the  pin  is  given. 


Limiting  Numbers  of  Teeth. 


D. 

F. 

Minimum   Value, 

TEETH. 

PINS. 

6.44 

00 

INSIDE  GEAR. 

7 

89 

D. 

F. 

D. 

F. 

8 

32 

TEETH. 

PINS. 

9 

21 

4 

2 

10 

16 

3 

4 

16 

3 

11 

14 

4 

8 

36 

4 

12 

12 

5 

14 

TEETH. 

PINS. 

14 

11 

6 

35 

16 

10 

18 

9 

ANNULAR. 

MAX.  VALUES. 

21 

8 

27 

7 

37 

• 

PIN  GEARING. 

110 

5 

Tooth  =  Space. 

GO 

4.64 

PIN    GEARING. 

Limiting  Numbers  of  Teeth. 


217 


D. 

TEETH. 

F. 

PINS. 

D. 

TEETH. 

F. 

PLANES. 

D. 

TEETH. 

F. 

PINS. 

2.68 

00 

2.68 

00 

2 

9 

3 

27 

3 

54 

4 

11 

4 

21 

F. 

D. 

PINS. 

TEETH. 

5 

8 

5 

13 

6 

6 

6 

12 

3 

25 

8 

5 

7 

11 

2 

4 

16 

4 

Q 

10 

oo 

3.29 

11 

9 

. 

16 
53 

8 

7 

INSIDE  GEAR. 
MAX.  VALVES. 

EECESS  =  PITCH. 

PIN  =  POINT. 

00 

6.58 

336.  The  Path  of  Contact  in  Pin  Gearing. — In  the  elementary  form, 
the  pin  being  a  mathematical  point  in  the  circumference  of  the  fol- 
lower's pitch  circle,  that  circumference  is  itself  the  locus  of  contact. 

When  the  pin  has  sensible  diameter,  its  centre  yet  lies  always  in 
that  circumference  ;  the  common  normals  to  the  pin  and  its  driving 
tooth  are  chords  of  the  circle,  all  passing  through  the  point  A,  Fig. 
208.  Supposing  the  driver  to  turn  to  the  right,  then,  we  have  only 
to  set  off  on  each  of  these  chords  a  distance  equal  to  the  radius  of  the 
pin,  measured  from  the  circumference  toward  the  left  :  the  line  join- 
ing the  points  thus  located,  which  is  the  curve  called  the  limacon,  is 
the  path  of  contact. 

This  might  also  be  constructed  from  the  outline  of  the  working 
tooth,  by  the  process  of  Fig.  164  ;  but  the  method  above  described  is 
more  convenient ;  and,  as  will  be  found  more  fully  set  forth  in  the 


218 


PRACTICAL  POINTS — NOISE  AND   VIBRATION. 


Appendix,  it  involves  principles  which  enable  us  to  determine  with 
greater  accuracy  than  would  otherwise  he  possible,  certain  critical 
points  not  only  in  the  path  of  contact  itself,  but  in  the  tooth  outline 
also ;  since  the  latter,  evidently,  may  by  reversing  the  construction 
be  derived  from  the  limacon. 

Some  Practical  Considerations. 

337.  Noise  and  Vibration. — Although  not  coming  strictly  within  the 
scope  of  this  treatise,  the  practical  ill  effects  of  using  wheels  with 
incorrectly  shaped  teeth  are  so  closely  connected  with  the  subject  as 
to  demand  a  brief  notice. 

It  is  to  be  observed,  then,  that  the  noise  and  the  vibration  which 
often  attend  the  action  of  toothed  gearing,  especially  at  high  speeds,  arc 

not  necessarily  identical  in  origin. 
Jt  is  true  that  the  causes  which 
produce  noise  will  also  produce  vi- 
bration ;  but  vibration  may  be  pro- 
duced by  other  causes,  and  may  at 
least  be  imagined  to  occur  without 
noise. 

To  explain;  suppose  two  wheels 
of  perfect  form  and  finish  to  gear 
with  each  other,  the  power  and  the 
resistance  being  absolutely  uniform; 
•then,  whatever  the  amount  of  back- 
lash, there  would  be  neither  vibra- 
tion nor  noise.  Now  in  practice, 
this  uniformity  of  power  and  re- 
208.  sistance  seldom  or  never  exists,  and 

the  variations  in  speed  cause  the  fronts,  and  often  the  backs,  of  the 
teeth  to  strike  together  at  short  intervals.  These  blows  cause  a  rat- 
tling noise,  which  is  worse  the  higher  the  speed,  and  is  accom- 
panied by  vibrations  due  to  the  impact  between  the  teeth. 

The  sole  cause  of  the  noise,  evidently,  is  the  existence  of  backlash ; 
but  even  were  the  teeth  so  perfect  as  to  have  no  backlash  at  all,  these 
irregularities  in  the  power  and  the  resistance  would  still  give  rise  to 
vibrations,  more  or  less  injurious  according  to  the  suddenness  of  the 
changes ;  they  would,  however,  take  place  in  quiet. 

•338.  But  again,  vibratory  action  may  result  from  a  totally  different 
cause,  namely,  incorrect  forms  of  the  teeth. 

To  illustrate  this,  imagine  two  engaging  wheels,  whose  teeth  are,  as 
before,  of  perfect  finish,  but  not  of  proper  contour  ;  let  the  speed  of 


EFFECTS   OF   WEAR.  219 

the  driver  be  absolutely  uniform,  and  the  resistance  such  as  to  keep 
the  acting  outlines  always  in  contact,  so  that  there  is  none  of  the  rat- 
tling above  mentioned. 

The  average  velocity  ratio  will  be  correct ;  if  the  driver  has,  for 
instance,  100  teeth  and  the  follower  50,  each  revolution  of  the  former 
will  cause  two  revolutions  of  the  latter.  And  further,  TJ¥  of  a  revo- 
lution of  the  driver  will  cause  -^  of  a  revolution  of  the  follower  ;  but 
during  this  fractional  motion  the  velocity  ratio  is  not  constant, 
the  follower  being  driven  too  rapidly  during  one  part  of  the  action, 
and  too  slowly  during  the  other  part. 

Thus  the  action  of  each  pair  of  teeth,  though  correct  as  a  whole,  is 
faulty  in  detail,  being  made  up  of  two  counterbalancing  errors.  The 
speed  of  the  driver  is  uniform,  but  that  of  the  follower  is  fluctuating  ; 
its  motion  consists  of  a  series  of  pulsations,  not  necessarily  audible 
at  low  speeds,  though  practically  certain  to  become  so  at  high  ones. 
But  even  at  moderate  velocities,  this  vibration  acts  injuriously  upon 
the  whole  mechanism  ;  and  in  many  cases  it  is  easy  to  see  that  the 
perfection  of  the  work  done  by  the  machine  may  be  impaired  by  the 
irregularity  of  its  motion,  no  matter  how  slowly  it  runs. 

339.  Effect  of  Wear  upon  Teeth  of  Incorrect  Forms. — It  is  clearly 
poor  economy  to  require  a  machine  to  finish  itself,  even  were  it  always 
able  and  willing  to  do  so,  since  in  the  meantime  its  proper  work  must 
be  imperfectly  executed.  Still  it  is  sometimes  asserted  that  teeth 
originally  of  incorrect  contour  will  "wear  into  shape"  and  improve 
by  use. 

Now  wear  means  abrasion ;  so  that  the  only  errors  which  could 
thus  correct  themselves,  by  the  removal  of  superfluous  material,  must 
be  errors  of  excess,  not  of  deficiency.  But  these  are  the  very  errors 
which  would  hardly  be  considered  admissible  by  the  most  zealous 
defender  of  bad  workmanship,  as  a  little  reflection  will  show.  Let 
us  suppose  that  of  two  wheels,  accurately  spaced,  the  faces  only  of  the 
driver's  teeth  are  too  full,  all  the  other  curves  being  correct.  Under 
these  circumstances,  it  will  be  seen  that  during  the  receding  action, 
the  leading  tooth  of  the  driver  will  urge  the  follower  too  rapidly ;  in 
order  to  admit  of  this,  then,  considerable  backlash  is  necessary.  A 
little  space  will  be  thus  left  between  what  should  be  the  acting  curves 
of  the  following  pair ;  which  cannot  come  into  action  until  the  lead- 
ing tooth  quits  contact,  allowing  the  follower  for  an  instant  to  wait 
for  the  driver  to  overtake  it.  Therefore,  in  addition  to  the  disadvan- 
tage of  having  all  the  work  done  by  one  tooth  at  a  time,  there  will  be 
a  constant  clattering  as  the  teeth  come  successively  into  driving 
contact. 


220  EFFECTS   OF   WEAR. 

If,  on  the  other  hand,  the  faces  of  the  follower  only  are  too  full ;  it 
is  then  the  last  instead  of  the  leading  tooth  which  will  have  the  work 
to  do,  and  the  follower  will  be  driven  beyond  the  normal  speed  during 
the  approach. 

But  it  will  be  driven  most  rapidly  during  the  first  part  of  the  ap- 
proach, its  speed  diminishing  as  the  point  of  contact  nears  the  line  of 
centres  :  and  though  there  will  not  be  an  absolute  halt  or  interruption 
of  the  action,  as  in  the  previous  case,  there  will  still  be  something 
like  a  blow  as  each  tooth  of  the  driver  comes  into  contact,  and  the 
wheels  will  be  nearly  as  noisy  as  before. 

Like  reasoning  applies  if  the  faces  be  made  correct  and  the  flanks 
too  full ;  so  that  practically  the  only  errors  in  the  forms  of  the  teeth 
which  the  worst  of  workmen  would  admit  in  the  rudest  of  mechanism, 
are  errors  of  deficiency.  These  may  cause  less  noise,  but  they  will 
certainly  cause  the  pulsation  above  described ;  and  when  they  correct 
themselves  by  wear,  we  shall  be  called  upon  to  contemplate  the  phe- 
nomenon of  growth  by  abrasion . 

340.  Wearing  to   a  Bearing. — The    fallacy   above    mentioned,   if 
crushed  by  theory,  is  pulverized  by  practice  ;  for  we  believe  that  the 
first  case  has  yet  to  be  found,  where  teeth  whose  original  contours  were 
not  full  enough,  have  worn  into  forms  which  made  the  velocity  ratio 
more  nearly  constant ;  but  every  repair   shop  can  furnish  evidence 
that  in  this  respect  such  teeth  have  gone  from  bad  to  worse. 

Nevertheless,  it  cannot  be  disputed  that  gear  wheels,  both  cast  and 
cut,  very  frequently  run  better  after  being  a  while  in  use.  In  the 
case  of  cast  wheels,  this  may  be  due  to  the  removal  by  abrasion  of 
rough  and  irregular  spots  in  the  metal ;  but  in  both  cases  it  may  be 
due  to  other  causes.  It  is  likely  to  happen,  especially  if  the  wheels 
be  of  considerable  thickness,  that,  owing  to  some  slight  error  either 
in  the  process  of  cutting  or  in  the  keying  of  the  wheel  upon  the  shaft, 
some  of  the  elements  of  the  teeth  are  not  exactly  parallel  to  the  axis. 
And  what  is  technically  called  "  wearing  to  a  bearing,"  resulting  in  a 
general  improvement  of  the  action  in  smoothness  if  not  in  constancy 
of  the  velocity  ratio,  consists  in  point  of  fact  not  in  the  accretion  of 
metal,  where  a  deficiency  exists,  but  in  the  removal  of  so  much  as 
stands  in  the  way  of  line  contact  between  the  teeth  ;  and  the  gain  is 
mainly  due  to  a  more  equal  distribution  of  the  pressure. 

341.  Effects  of  Wear  in  the  Bearing. — In  order  to  preserve  a  constant 
velocity  ratio,  the  teeth  of  a  pair  of  wheels  must  be  conjugate  to  each 
other  ;  and  if  they  be  of  other  than  involute  form,  it  is  also  necessary 
that  the  distance  between  centres  shall  be  equal  to  the  sum  of  the 
radii  of  the  pitch  circles.     But  unless  special  provision  is  made  for 


EFFECTS   OF   WEAR   IN   THE   BEARINGS. 


221 


readjustment,  which  cannot  always  bo  done,  this  distance  is  subject 
to  variation  on  account  of  wear  in  the  bearings,  so  that  the  wheels 
after  a  time  will  either  mesh  too  deeply,  or  not  deeply  enough  ;  and 
it  is  easy  to  see  that  if  the  line  of  centres  be  thus  made  too  short  or 
too  long  by  a  given  amount,  the  effect  upon  the  velocity  ratio  may  be 
greater  with  one  tooth-system  than  with  another,  although  both  may 
originally  work  with  perfect  precision. 

It  will  subsequently  be  shown  that  it  is  just  as  feasible  to  give  a 
practical  embodiment  to  Sang's  theory  as  to  the  epicycloidal  system, 
and  to  produce  accurate  templates  for  interchangeable  wheels,  derived 
from  any  assumed  rack,  by  automatic  machinery.  When  this  shall 
have  been  done,  the  question  will  naturally  arise,  which  of  the  various 


MAC  CORD'S  ODONTOSCOPE.    FIG.  209. 

tooth-systems  thus  placed  within  equally  easy  reach  will  be  affected 
to  the  least  extent  by  the  disturbing  causes  above  pointed  out ;  for 
the  one  which  is,  will  clearly  be  the  best  for  many  purposes. 

342.  The  Odontoscope. — In  order  to  answer  this  question,  it  is  neces- 
sary to  have  some  convenient  means  of  comparing  the  actions  of  dif- 
ferent tooth-systems  under  varying  conditions.  Neither  graphical 
nor  analytical  treatment  appearing  eligible  on  account  of  the  exceed- 
ingly intricate  nature  of  the  problems  involved,  the  author  some  years 
since  devised  the  apparatus  of  which  the  principle  is  illustrated  in 
Fig.  209.  This  was  before  the  introduction  of  automatic  machinery 
for  the  shaping  of  gear  cutters,  and  the  original  object  was  to  test  the 
accuracy  of  such  as  were  made  by  hand. 


222  MAC   COED'S   ODCXNTOSCOPE. 

For  this  purpose,  two  templates,  A  and  B,  are  cut  out  of  tolerably 
thick  sheet  metal,  corresponding  to  the  teeth  of  a  pair  of  wheels ; 
these  are  secured  to  arms  turning  about  the  axes  C  and  2),  whose 
distance  from  each  other  is  capable  of  adjustment.  The  shaft  of  D 
carries  a  graduated  limb,  K,  and  may  be  slowly  rotated  by  means  of 
the  arm,  L,  and  tangent  screw,  E,  or  any  equivalent  'device  ;  motion 
will  thus  be  communicated  to  C,  the  tooth  A  being  kept  in  contact 
with  B,  by  a  light  weight  or  spring,  not  shown. 

J^and  G  are  small  cylindrical  barrels,  accurately  turned  to  diameters 
having  the  same  ratio  as  those  of  the  pitch  circles.  F  is  fixed  on  the 
axis  (7,  while  ft,  which  carries  a  pointer,  H,  turns  freely  on  the  axis 
D,  with  which  it  is  connected  by  a  spring,  not  shown  ;  the  tendency 
being  to  wind  up  on  G  the  fine  flexible  wire  /,  secured  to  both  bar- 
rels in  the  manner  of  a  crossed  belt.  Thus  F  and  G  will  always  turn 
in  opposite  directions,  with  a  velocity  ratio  perfectly  constant  from 
the  nature  of  the  connection.  The  velocity  ratio  of  the  motions  of 
C  and  D,  however,  is  determined  by  the  templates  A  and  B,  and  will 
not  remain  constant  unless  their  contours  are  strictly  conjugate. 

The  pitch  circles  being  scribed  on  the  templates,  it  will  be  apparent 
that  when  the  intersections  of  these  circles  with  the  contours  of  the 
teeth  are  brought  into  coincidence,  the  centres  of  motion  are  at  the 
correct  distance  from  each  other,  and  the  point  of  contact  is  also 
exactly  on  the  line  CD.  By  turning  the  tangent  screw  in  one  direc- 
tion or  the  other,  then,  we  may  examine  the  action  during  the  arc  of 
approach  or  that  of  recess  at  pleasure  ;  if  the  templates  are  correctly 
shaped,  the  limb  K  and  the  pointer  ^Twill  move  at  the  same  rate  and 
in  the  same  direction,  so  that  if  the  latter  be  set  at  zero  on  the  grad- 
uated arc,  it  will  remain  at  zero  throughout  the  action  ;  any  inac- 
curacy being  manifested  by  "a  movement  of  the  pointer  over  the  limb. 

343.  The  sensitiveness  of  the  instrument  as  designed  for  actual 
work  is  increased  by  the  introduction  of  multiplying  gear  between 
the  barrel  or  pulley  G  and  the  pointer  H\  thus  producing  a  greater 
deflection  for  an  error  of  given  magnitude,  and  causing  a  very 
minute  one  to  proclaim  its  existence.  Its  exact  locality,  however,  we 
can  not  fix ;  that,  indeed,  would  be  equivalent  to  determining  the 
point  of  contact  of  two  tangent  curves  ;  and  if  the  end  in  view  is  the 
correction  of  errors  in  the  contours  of  the  templates,  the  only  re- 
source is  careful  inspection  with  a  magnifying  glass. 

But  for  comparative  work  like  that  previously  alluded  to,  this  in- 
strument is  well  adapted.  Suppose  that  the  templates  work  with 
perfect  accuracy  when  the  centres  are  correctly  adjusted ;  by  means 
of  a  screw,  not  shown,  the  bearing  of  G  may  now  be  moved  either  to 


TEETH   OF   NO^-CIRCULAR   WHEELS. 


223 


the  right  or  the  left  through  any  given  distance  :  then  let  the  deflec- 
tion of  the  pointer  be  noted  for,  say  every  degree  or  half  degree  of  the 
angle  of  action.  If  this  be  repeated  after  other  changes  in  the  distance 
CD,  a  comparison  of  these  records  with  similar  ones  for  a  different 
pair  of  templates,  will  furnish  all  needed  information  as  to  the  effects 
of  wear  in  the  bearings  upon  the  velocity  ratio  in  the  two  systems  of 
gearing  represented. 

Teeth  of  Non- Circular  Wheels. 

344.  When  two  pitch  curves  of  any  form  roll  together  about  fixed 
axes,  the  point  of  contact  always  lies  upon  the  line  of  centres,  and 
divides  it  into  segments  inversely  proportional  to  the  angular  velocities 
at  the  instant. 

Through  this  point  there  must  pass 
the  common  normal  of  any  two  con- 
jugate tooth-curves  ;  these,  conse- 
quently, must  satisfy  the  condition  of 
(181),  viz.:  they  can  be  traced  upon 
the  planes  of  rotation  during  the 
action,  by  a  marking  point  invariably 
connected  with  a  describing  line  which 
moves  in  rolling  contact  with  both 
pitch  curves. 

The  forms  of  non-circular  wheels 
are  so  numerous  and  diverse,  that  ifc 
would  be  an  endless  task  to  discuss  at 
length  the  various  possibilities  rela- 
ting to  the  paths  of  contact,  limiting  FIG.  210. 
numbers  of  teeth,  and  matters  of  kindred  nature. 

But  the  use  of  such  wheels  is  so  limited  that  the  consideration  of 
all  these  questions  has  by  no  means  the  importance  or  interest  which 
attach  to  it  in  relation  to  ordinary  gearing  :  and  we  shall  accordingly 
be  content  with  briefly  explaining  a  purely  practical  method  of  con- 
structing the  teeth  by  graphic  operations. 

345.  Laying  out  the  Teeth  of  Elliptical  Wheels,— In  Fig.  210,  let  AO 
be  the  semi-major  and  10  the  semi-minor  axis  of  a  pitch  ellipse. 
Although  no  part  of  this  curve  is  truly  circular,  yet  circular  arcs  may 
be  found  which  agree  with  it  so  closely  that  for  all  practical  purposes 
they  may  be  substituted  for  it. 

Thus  by  trial  and  error  we  find  a  centre,  (7,  on  the  major  axis,  such 
that  an  arc  described  about  it  will  coincide  with  the  ellipse  in  the 
region  of  the  vertex  A,  so  nearly  that  the  difference  will  not  be  per- 


224  ELLIPTICAL   GEARING—  FORMS    OF   TEETH. 

ceptiblc,  even  when  the  curve  is  drawn  with  the  finest  visible  line. 
The  deviation  will,  however,  become  apparent  if  the  arc  be  extended, 
and  it  should  be  terminated  at  some  point,  B,  a  little  before  this  limit 
has  been  reached.  If  the  closest  possible  approximation  is  desired,  a 
normal  through  B  should  pass  througli  C ;  if  on  trial  this  prove  not 
to  be  the  case,  a  minute  change  in  the  position  of  either  B  or  (7,  or 
both  if  necessary,  will  cause  this  condition  to  be  satisfied.  On  BO 
produced,  we  now  seek  for  another  centre  D,  such  that  an  arc  BE 
described  about  it  through  B  shall  sensibly  coincide  with  the  contin- 
uation of  the  elliptical  contour.  Proceeding  in  this  way,  we  shall 
finally  have  a  centre,  H,  upon  the  minor  axis,  for  the  arc  GI,  which 
completes  the  quadrant  of  the  ellipse. 

For  the  purpose  of  illustration,  we  have  in  this  diagram  represented 
an  ellipse  whose  eccentricity  is  much  greater  than  that  of  any  likely 
ever  to  be  used  as  the  pitch  curyc  of  a  wheel ;  yet  a  very  perfect  ap- 
proximation is  made  by  means  of  the  four  circular  arcs  for  each 
quadrant ;  probably  three  will  suffice  for  any  case  to  be  met  with  in 
practice,  and  in  most  instances  two  answer  the  purpose  abundantly 
well. 

346.  The  next  step  is  to  subdivide  the  perimeter  of  the  ellipse  for 
the  location  of  the  teeth  and  spaces.     The  outline  now  being  com- 
posed of  circular  arcs,  these  are  rectified  by  Prof.  Rankine's  graphic 
process.     Thus  in  the  figure,  we  lay  off  upon  the  tangent  at  G,  the 
lengths  GE'9  GT  of  the  arcs  GE,  GI\  similarly  upon  the  tangent  at 
By  the  lengths  of  BE  and  BA  are  laid  off,  and  their  sum  being  added 
to  I'E'  already  found,  gives  I' A'  equal  in  length  to  the  quadrant 
IEA.     This  right  line  is  next  to  be  properly  subdivided  according  to 
the  number  of  teeth  determined  on,  and  the  points  of  division  are 
then  transferred  to  the  ellipse  by  means  of  Prof.  Rankine's  converse 
process  applied  to  the  approximating  arcs  and  their  rectifications  : 
this  spacing  is  indicated  in  the  diagram  by  the  alternating  fine  and 
heavy  lines  on  both  tangents  and  contour. 

A  circle  is  used  for  generating  the  tooth-outlines,  rolling  it  first 
within  the  ellipse  to  trace  the  flanks,  and  then  without  to  describe  the 
faces ;  and  thus  this  operation  is  made  identical  with  that  for  circu- 
lar wheels.  The  radius,  AL,  of  this  describing  circle  should  not  be 
greater  than  -J-  AC,  which  limiting  value  will  give  radial  flanks  to  the 
teeth  whose  edges  cut  the  ellipse  upon  the  approximating  arc  AB ; 
the  other  flanks  being  concave. 

347.  The  complete  wheels  in  gear  are  represented  in  Fig.  211.     In 
regard  to  the  number  of  teeth,  it  makes  no  difference  whether  it  be 
even  or  odd  ;  but  in  either  case  it  is  eminently  desirable  that  the  two 


ELLIPTICAL   GEARING ARRANGEMENT   OF   TEETH. 


225 


wheels  should  be  alike.     In  order  to  make  them  so,  if  there  be  an  odd 

number  of  teeth,  they  should  be  so  arranged  that  one  extremity  of 

either  the  major  or  the  minor 

axis  shall   bisect  a  tooth,  the 

opposite  extremity  bisecting  a 

space.     If  the  number  of  teeth  > //\  h 

be  even,  then  they  must  be  so 

placed  that  the  fronts  of  two  of 

them  shall  cut  the  pitch  ellipse 

at  opposite  extremities  of  one  of 

the  axes. 

Now,  the  action  of  the  teeth 
on  different  parts  of  the  peri-  FIQ.  211. 

meter  cannot,  under  any  circumstances,  be  equalized  in  all  particu- 
lars. In  order  to  secure  the  same  amounts  of  approaching  and  reced- 
ing action,  it  would  be  necessary  to  make  the  faces  of  varying  lengths  ; 
and  even  then  the  result  is  of  doubtful  utility,  cincc  the  degree  of 
obliquity  is  continually  changing.  Consequently  it  is  held  to  bo  suf- 
ficient in  practice  to  make  the.  faces  of  uniform  length ;  and  this 
should  be  such  as  to  ensure  that  in  the  part  of  the  whole  movement 
in  which  the  obliquity  of  action  is  greatest,  one  pair  of  teeth  shall  not 
quit  contact  until  the  next  pair  has  fairly  come  into  engagement. 
Thus  the  form  of  the  blank  is  determined  by  drawing  a  curve  par- 
allel to  the  pitch  ellipse  at  a  given  distance  without,  the  bottoms  of 
the  spaces  are  bounded  by  another  at  a  given  distance  within,  and  the 
two  blanks,  secured  side  by  side,  may  be  cut  at  the  same  time  with 

milling  cutters  precisely  like  those  used  for 
circular  wheels. 

It  is  not  necessary  to  furnish  elliptical 
wheels  with   teeth  all  round  their  periph- 
eries, when  used  under  circumstances  which 
permit  their  free  foci,  as  in  Fig.  212,  to  be 
connected  by  a  link.     In  this  case,  a  few 
teeth  near  the  extremities  of  the  major  axis, 
as  shown,  suffice  to  carry  the  wheels,  past 
the  dead  points,  which  occur  when  the  line 
of  the  link  coincides  with  the  line  of  cen- 
tres.   The  remaining  portions  of  each  wheel 
must  be  formed  exactly  to  the  contour  of 
the  pitch  ellipse  ;  these  will  then  transmit  the  rotation  by  direct  and 
purely  rolling  contact,  in  either  direction,  so  long  as  the  contact 
radius  of  the  driver  is  on  the  increase,  (88).     Since  while  the  radius 
15 


FIG.  212. 


226  ELLIPTICAL   WHEELS — INVOLUTE   TEETH. 

is  on  the  decrease,  the  work  must  be  done  by  the  link,  it  is  necessary 
to  use  a  sufficient  number  of  teeth  to  prevent  excessive  obliquity  of 
action  while  the  link  is  thus  made  the  sole  means  of  transmission. 

348.  Elliptical  Wheels  with  Involute  Teeth. — It  is  worthy  of  notice, 
though  more  as  a  matter  of  abstract  interest  than  cf  practical  mo- 
ment, that  the  teeth  for  a  pair  of  pitch  ellipses  may  be  made  in  the 
form  of  involutes  of  smaller  base  ellipses  having  the  same  foci.  In 
Fig.  213,  let  0  be  the  fixed  and  A  the  free  focus  of  the  ellipse  whose 
major  axis  is  XX.  Let  J9,  at  any  distance  from  C,  be  the  fixed  focus 
of  an  equal  and  similar  ellipse,  the  position  of  the  major  axis  YY 
being  found  as  follows  :  about  D  describe  an  arc  with  radius  =  CA, 
and  about  A  an  arc  with  radius  —  CD  ;  these  arcs  intersect  at  B, 


FIG.  213. 

which  is  the  free  focus  of  the  second  ellipse.  Draw  CD  and  AB, 
intersecting  at  P.  The  common  tangent  to  the  two  ellipses,  Off, 
will  also  pass  through  P  (see  Appendix) ;  and  if  we  suppose  it  to 
represent  an  inextensible  band,  wrapped  around  and  secured  to  the 
ellipses,  it  will  be  wound  upon  the  right  hand  one  if  it  turn  about 
C  as  shown  by  the  arrow,  and  unwound  from  the  other,  which  will 
thus  be  caused  to  revolve  about  D,  with  a  definite  varying  velocity 
ratio.  During  this  motion,  a  marking-point  fixed  at  P  in  this  band, 
and  moving  with  it,  will  trace  upon  the  planes  of  rotation  the  invo- 
lutes of  the  ellipses,  KPL,  OPL  These  curves,  therefore,  if  used  as 
the  outlines  of  teeth,  will  maintain  the  same  velocity  ratio  that  was 
originally  established  by  the  band. 

349.  But,  as  will  be  seen  by  comparing  this  diagram  with  Fisr.  55, 


ELLIPTICAL   WHEELS — INVOLUTE   TEETH.  227 

P  is  also  the  common  point,  and  TPT  is  the  common  tangent,  of  two 
ellipses  confocal  with  the  original  ones,  whose  major  axes  are  WV, 
UZ,  equal  to  each  other  and  to  CD ;  and  these  will  roll  in  contact 
about  the  fixed  foci. 

Thus  four  different  elementary  combinations  are  represented  in  this 
figure  ;  we  may  use  either 

1.  The  levers  CA,  DB,  connected  by  the  link  AB  ; 

2.  The  smaller  ellipses,  connected  by  the  band  GH\ 

3.  The  involutes  KL,  10,  acting  with  sliding  contact ;  or 

4.  The  larger  ellipses,  acting  in  pure  rolling  contact.     Or,  these 
may  be  used  simultaneously,  the  velocity  ratio  being  the  same  in  each 

GW    CV 
at  any  and  every  instant,  and  varying  between  the  limits  ~7Ty9  ~7rwf- 

350.  The  distance  CD  being  arbitrary,  it  follows  that  for  the  same 
base  ellipses,  an  infinite  number  of  pitch  ellipses  may  be  assigned. 
And  on  the  other  hand,  for  the  same  pitch  ellipses,  the  teeth  may  be 
involutes  of  any  base  ellipses  having  the  same  foci.  From  which  the 
deduction  is,  that  so  long  as  the  teeth  of  this  form  engage  at  all,  they 
will  gear  correctly  in  the  sense  that  the  action  will  be  equivalent  to 
the  rolling  together  of  a  pair  of  true  ellipses  of  some  form  :  but 
whereas  in  the  case  of  circular  wheels  the  velocity  ratio  was  not 
affected  by  altering  the  distance  between  the  centres,  it  is  to  be  noted 
that  in  elliptical  gearing  the  limits  between  which  the  velocity  ratio 
varies,  will  be  affected  by  such  alteration.  For  the  focal  distance  CA 
is  invariable,  while  the  major  axis  WV  is  always  equal  to  CD ;  so 
that  a  change  in  the  length  of  the  line  of  centres  changes  the  eccen- 

CV 

tricity  of  the  pitch  ellipse  and  the  value  of  — — r . 

If  then  it  be  essential  that  the  limits  of  variation  should  be  exactly 
maintained,  the  involute  form  of  tooth  possesses  no  advantage  over 
that  previously  described  ;  and  it  is  hardly  necessary  to  add  that  if 
the  free  foci  of  a  pair  of  elliptical  wheels  which  arc  provided  with 
teeth,  be  connected  by  ti  link,  the  distance  between  the  fixed  foci  must 
be  kept  constant,  on  whatever  system  the  teeth  are  laid  out.  If  the 
involute  system  be  adopted,  care  must  be  taken  lest  the  base  ellipses 
be  so  small  as  to  produce  excessive  obliquity  of  action.  In  regard  to 
this,  Prof.  Kankine,  to  whom  is  due  the  credit  of  first  discussing  this 
form  of  tooth  for  elliptical  gearing,  observes  that  the  common  tan- 
gent should  cut  the  fronts  of  at  least  two  teeth  between  the  base  and 
the  pitch  ellipse  ;  which,  although  it  may  not  be  necessary  in  all  cases 
in  order  to  ensure  the  transmission  of  rotation,  is  unquestionably  a 
safs  practical  rule. 


228 


TEETH   OF   LOBED   WHEELS. 


351.  Construction  of  Teeth  of  Lobed  Wheels, — As  above  intimated,  the 
method  of  operation  first  described  is  perfectly  general,  and  may  be 
employed  with  pitch  curves  of  any  form.     The  centres  of  the  circular 
arcs  of  which  the  contour  is  to  be  made  up,  may  be  found  by  assum- 
ing a  number  of  points,  at  which  the  curvature  begins  sensibly  to 
change,  and  drawing  normals  through  these  points  (graphic  processes 
for  which  are  described  in  the  Appendix)  ;  the  intersections  of  the 
normals  being  the  required  centres. 

But  if  the  pitch  curves  be  carefully  drawn,  it  will  be  found  that  for 
all  practical  purposes,  these  centres  can  be  located  with  sufficient 
accuracy  by  trial  and  error ;  and  at  best  the  laying  out  of  teeth  for 
such  wheels  requires  much  time,  care,  and  patience. 

If  the  same  describing  circle  be  used  throughout,  its  diameter 
should  be  such  as  to  give  radial  flanks  to  the  teeth  in  that  part  of  the 
pitch  line  where  the  curvature  is  greatest ;  should  other  parts  be  very 
much  flatter  the  teeth  may  in  consequence  spread  too  rapidly  at  the 
root  or  in  the  flank.  This  may  be  remedied  by  using  different  de- 
scribing circles  for  the  teetli  in  those  parts,  care  being  taken  that  the 
same  one  be  always  used  for  the  conjugate  face  and  flank. 

352.  Practical  Limit  to  the  Obliquity.— In  any  form  of  gearing  it 
would  be  very  desirable  to  have  the  line  of  action  perpendicular  to 


FIG.  214. 

the  line  of  centres,  since  then  there  would  be  no  component  of  side 
pressure. 

In  speaking  of  the  teeth  of  circular  wheels,  the  obliquity  was  de- 
fined as  the  angle  made  by  the  common  normal  of  the  acting  curves 
with  the  common  tangent  to  the  pitch  circles.  This,  however,  was 
because  this  tangent  is  then  a  perpendicular  to  thfe  line  of  centres; 


LOBED    WHEELS — LIMIT   OF   OBLIQUITY. 


229 


and  in  general  the  obliquity  is  the  inclination  of  the  line  of  action  to 
such  a  perpendicular,  which  latter  may  or  may  not  coincide  with  the 
common  tangent  to  the  pitch  curves.  This  it  seldom  does  in  the 
class  of  wheels  now  under  consideration,  and  in  consequence  a  much 
greater  obliquity  than  would  be  admissible  in  circular  gearing  is  often 
unavoidable. 

Thus  in  Fig.  214,  VW,  XY  are  arcs  of  the  pitch  curves,  in  con- 
tact at  P,  on  the  line  of  centres  CD,  to  which  PO  is  perpendicular. 
The  lower  wheel  driving,  as  shown  by  the  arrow,  let  E  be  the  point 
of  tooth-contact,  the  line  of  action  through  which,  PEN,  has  the 
greatest  inclination  to  PT,  the  common  tangent  to  the  pitch  curves. 

It  will  then  be  apparent 
that  when  the  angle  TPO, 
which  this  tangent  makes 
with  the  perpendicular  PO 
is  greatest,  the  total  obliquity 
NPO  will  be  a  maximum. 

The  results  of  experience 
show  that  the  angle  TPO  FIG.  215. 

should  not,  if  it  be  practicable  to  avoid  it,  exceed  from  25°  to  30°. 
In  other  words,  the  contours  of  the  non-circular  pitch  curves  should 
be  such  that  the  angle  between  the  tangent  and  the  radius  vector 
shall  at  no  point  be  less  than  60°  or  65°.  Nor  should  the  angle  NPT 
exceed  from  15°  to  20° ;  thus  limiting  the  range  of  the  maximum 
total  obliquity,  NPO,  to  from  40°  to  50°;  though  if  the  object  be 
rather  the  modification  of  motion  than  the  performance  of  heavy 
work,  higher  values  may  in  extreme  cases  be  used. 

353.  This  limit,  it  will  be  perceived,  has  not  been  regarded  in  Fig. 
215,  which  shows  the  appearance  of  the  dissimilar  pitch  curves  of  Fig. 
75,  when  furnished  with  teeth. 

In  Fig.  216,  we  have  a  pair  of  similar  but  unsymmetrical  uni- 
lobes,  the  pitch  lines  being  elliptical  below  the  horizontal  centre 
line,  and  of  the  logarithmic  spiral  form  above,  as  in  Fig.  72. 

The  lower  halves  of  the 
dissimilar  pair  shown  in 
Fig.  217  are  alike,  the  log- 
arithmic spiral  being  the 
pitch  curve ;  while  above 
the  line  of  centres,  that 
spiral  is  used  for  a  part  of 
FlG>  216<  the  contour,  the  remainder 

being  elliptical,  as  in  Fig.  74.     But  the  wheels  here  represented  pre- 


230 


LOBED   WHEELS— VARIOUS   EXAMPLES. 


sent  the  peculiarity  that  the  limits  of  variation  in  the  velocity  ratio, 

which  suddenly  change  when  the  wheels  reach  the  position  given  in 

the  figure,  are  not  the 
same  for  the  two  halves 
of  the  revolution. 

A  still  greater  and 
more  abrupt  transition 
is  produced  by  the  pair 
represented  in  Fig.  218, 
of  which  the  pitch 
curves  are  similar  and 
equal  logarithmic 
spirals.  It  is  tolerably 
evident  that  in  this  case 
the  absolute  velocity 
must  be  small  in  order 
to  avoid  violent  shocks 
at  the  instants  of  the 
change  :  yet  wheels  very 
similar  to  these  have 

been  used — as  for  instance  in  the  moulding  machine  of  Gallas  and 

Aufderheide.* 

354.  Non-Circular  Pin 

Gearing. — The  labor  of 

laying  out  the  teeth  of 

a     pair     of     dissimilar 

wheels  may  be  abridged, 

and    the    difficulty    of 

cutting    them    in    part 

avoided,  by  making  one 

of  them  after  the  man- 
ner of  a  pin-wheel.    The 

principle  of  the  process 

(which  may  also  be  ap- 
plied to  the  ellipses  or 

to  any  similar  as  well  as 

to     dissimilar     lobed 

wheels)  is  precisely  the  FIG.  sis. 

same  as  in  the  case  of  circular  gearing.     That  is  to  say,  the  pitch 

curve,  of  the  follower  is  used  as  the  describing  line,  by  rolling  which 


*For  description,  see  "Mechanics"  of  February  18,  1882. 


PI^-GEARI^G   FOR   LOBED   WHEELS. 


231 


upon  the  pitch   curve  of  the  driver  are  generated  the  elementary 

teeth  for  the  latter  ;  and  the 

working  outlines    are    at  a 

constant     normal     distance 

within  these,   equal  to  the 

radius  of  the  pin,  as  shown 

in  Fig.  219. 

In  regard  to  the  size  of  the 
pins,  and  their  distance  from 
each  other,  it  is  sufficient  to 
say  that,  as  the  contour  of 
each  wheel  is  regarded  as 
made  up  of  approximating 
circular  arcs,  the  whole  op- 
eration is  reduced  to  the  fa- 
miliar one  of  constructing  epi- 
cycloids and  their  parallels; 
so  that  the  limits  in  relation 
to  the  pitch  and  the  diameter 
of  the  pins  are  readily  de- 
duced for  each  particular  case 
by  application  of  the  reason- 
ing given  in  the  discussion 
of  circular  gearing  of  this 
description.  Though  it  may  FIG.  219. 

be  well  to  repeat  that  the  teeth  in  all  cases  should  be  as  small  and  as 
numerous  as  they  can  be  made  consistently  with  securing  the  neces- 
sary strength. 


CHAPTER  XII. 


1.  THE  TEETH  OF  BEVEL  WHEELS. — General  Principles  of  their  Correct  Forma- 

tion by  means  of  a  Describing  Cone.  Tredgold's  Approximate  Method.  De- 
tails of  correct  Process.  Results  of  the  two  Methods  Compared,  for  Epicy- 
cloidal  Teeth.  Inside  Bevel  Gearing.  The  Involute  System.  Action  of 
Bevel  and  Spur  Wheels  compared.  Teeth  of  Conical  Lobed  Wheels.  Methods 
of  Cutting  the  Teeth.  Corliss's  Bevel  Gear  Cutter.  Twisted  Bevel  Wheels. 

2.  THE  TEETH  OF  SKEW  WHEELS. — Theory  of  Willis  and  Rankine.     Describing 

Hyperboloid.  Approximate  Methods.  Direct  Construction  of  Teeth  thus  Gen- 
crated.  Fallacy  of  this  Theory.  Such  Teeth  Impracticable.  A  New  Theory. 
Oliviers's  Involutes  in  Different  Planes.  The  Fronts  of  the  Teeth  Single 
Curved.  They  Vanish  at  the  Gorge.  The  Length  of  the  Tooth  Limited. 
Backs  of  Teeth  Warped.  The  Conjugate  Teeth  Unlike.  Action  Reversed  on 
Crossing  the  Gorge.  Twisted  Skew  Wheels. 

3.  THE  TEETH  OF  SCREW  WHEELS. — Common  Worm  and  Wheel.     Construction 

Referred  to  that  of  Rack  and  Pinion.  Distinctive  Features  of  the  Action. 
Close-fitting  Tangent  Screw.  Practical  Proportions.  Sang's  Theory  Embodied. 
Multiple- threaded  Screw  Wheels.  Screw  and  Rack.  Oblique  Screw  Gear- 
ing. Construction  of  Teeth  from  Oblique  Rack  and  Pinion.  Peculiar  modi- 
fication of  the  Action.  Close-fitting  Oblique  Worm.  Oblique  Screw  and  Rack. 
Construction  for  Least  Amount  of  Sliding.  Resemblance  to  Skew  Gearing. 
Hour-glass  Worm  Gear.  General  Arrangement.  Form  of  the  Pitch  Surface  of 
the  Worm.  Forms  of  the  Threads.  Action  Confined  to  one  Plane.  Forms 
of  the  Wheel-teeth.  Multiple- threaded  Hour-glass  Worm  with  Face-gear 
Wheel.  Rollers  Substituted  for  Teeth  upon  the  Wheel. 

4.  THE  TEETH  OF  FACE  WHEELS.— Equal  Wheels  with  Cylindrical  Pins.     Equal 

Wheels  with  Axes  Perpendicular  to  each  other.  Unequal  Wheels  Similarly 
Situated.  Unequal  Wheels  with  Axes  in  Different  Planes.  Miscellaneous 
Arrangements  of  Face  Wheels.  Combination  of  Face  and  Screw  Gearing. 
Spherical  Screw  and  Wheel. 

355.  In  Treating  cf  Spur  Wheels  it  has  been  convenient,  since  all 
the  transverse  sections  are  alike,  to  consider  all  the  motions  as  taking 
place  in  one  plane,  and  thus  to  deal  with  lines  instead  of  surfaces.  But 
we  must  not  lose  sight  of  the  fact  that  the  pitch  and  describing  curves, 
and  also  the  tooth-outlines,  are  but  the  bases  of  surfaces  with  elements 
perpendicular  to  the  paper,  and  acting  in  right-line  contact.  Thus 
in  the  epicycloidal  system,  for  instance,  we  must  imagine  a  describing 
cylinder  rolling  upon  the  pitch  cylinders,  the  common  element  being 


GENERATION   OF   THE   TOOTH   SURFACE. 


233 


UNIVERSITY 


FIG.  220. 


the  instantaneous  axis,  about  which  the  describing  line  is  revolving 

at  any  given  phase  of  the  action.      Now  the   teeth  thus  generated 

must  touch  each  other  along  a 

right  line:   and  this   principle 

is  capable  of  a  more  extended 

application.      The  very   slight 

modification    by    which     spur 

wheels    are    transformed     into 

bevel  wheels  is  at  once  suggested 

by  regarding  the  pitch  cylinder 

as  the  limiting  form  of  the  cone, 

in  which  the  vertex  is  infinitely 

remote  ;  and  this  leads  directly 

to  the  analogous  generation  of 

the  tookh  surface  by  the  rolling 

of  a  describing  cone  in  contact 

with  the  pitch  cones. 

This  is  illustrated  in  Fig.  220  ;  supposing  the  arcs  AP,  AS,  to 
be  equal,  then  while  the  smaller  cone  rolls  upon  the  larger,  the  ele- 
ment CP  will  generate  the  surface  CPB ;  to  which  the  plane  CPA  is 
normal,  since  at  the  phase  here  represented,  CA  is  the  instantaneous 
axis.  This  is  the  essential  property,  and  is  independent  of  the  forms 
of  the  bases  of  the  cones  ;  but  if  as  here  shown  these  be  circular,  the 
extremity  P  of  the  generating  radius  will  trace  a  curve,  BP,  lying  on 
the  surface  of  a  sphere,  and  called,  since  it  is  described  by  the  rolling 
of  one  circle  upon  another  (although  the  two  are  not  in  the  same 
plane),  a  spherical  epicycloid  ;  which  may  be  considered  as  the  direc- 
trix of  the  conical  surface  CPB. 

In  like  manner,  by  the  rolling  of  one  cone  inside  of  another,  both 
having  circular  bases,  a  spherical  hypocycloid  may  be  described.  And 
whatever  the  bases  of  the  cones,  it  is  evident  that  a  surface  is  gener- 
ated in  this  case  also,  to  which  the  plane  determined  by  the  describing 
line  and  the  element  of  contact  at  any  instant,  will  be  normal. 

It  is  thus  readily  seen  that,  the  describing  cone  being  tangent  ex- 
ternally to  one  pitch  cone  and  internally  to  the  other,  these  two  sur- 
faces will  be  swept  up  simultaneously  as  the  rotation  progresses,  and 
will  at  every  instant  have  a  common  normal  plane,  cutting  the  plane 
of  the  fixed  axes  of  rotation  always  in  the  common  element  of  the 
pitch  cones  :  they  are,  therefore,  correct  forms  of  tooth -surf aces,  and 
will  maintain  the  original  velocity  ratio,  be  the  same  constant  or 
variable. 

356.  Tredgold's  Approximation. — Before  considering  in  detail  the 


234 


APPROXIMATE  CONSTRUCTION  OF  TEETH. 


FIG.  221. 


practical  operation  of  laying  out  the  teeth  in  accordance  with  the 
exact  theory  as  above  set  forth,  it  is  proper  to  describe  the  method 
more  commonly  employed,  which  involves  possibly  a  little  less 

labor,  and  gives  results,  not 
rigidly  accurate,  but  suffi- 
ciently so  for  many  purposes 
under  ordinary  conditions. 

The  principle  of  this  proc- 
ess, which,  it  should  be  ob- 
served, is  applicable  only  in 
the  case  of  circular  wheels,  is 
shown  in  Fig.  221.  Let  CA, 
CB,  in  the  plane  of  the  paper, 
be  the  axes,  and  CP  the  com- 
mon element,  of  the  pitch 
cones  CPE,  CPF;  whose  bases  EP,  PF,  are  small  circles  of  the 
sphere  shown  in  dotted  outline.  Draw  A PB  perpendicular  to  CP; 
then  AP  by  revolving  around  CA  will  generate  the  cone  PAF,  and 
BP,  by  revolving  around  CB,  will  generate  the  cone  PBE :  these 
cones  are  tangent  to  the  sphere,  and  respectively  normal  to  the  pitch 
cones. 

Now,  AB  is  also  the  trace  of  a  plane  tangent  to  the  sphere  at  P, 
and  tangent  to  both  normal  cones  ;  and,  in  the  diagram  at  the  right, 
the  points  A',  P',  B',  are  the  projections  of  A,  P,  and  B,  upon  this 
tangent  plane,  and  the  arcs  OP'M,  LP'N,  are  the  developments  of 
the  bases  FP,  PE.  These  arcs  are  next  to  be  treated  as  though  they 
were  the  pitch  circles  of  spur  wheels,  and  teeth  are  to  be  laid  out 
upon  them  according  to  any  of  the  systems  previously  explained, 
that  of  pin-gearing  of  course  excepted. 

Suppose  these  teeth  to  be  cut  out  of  a  thin  sheet  of  metal,  and  then 
wrapped  back  upon  the  normal  cones  ;•  their  outlines  are  then  to  be 
traced,  and  treated  as  the  directrices  of  the  conical  tooth-surfaces,  all 
of  whose  elements,  as  before  explained,  converge  to  the  vertex  C  of 
the  pitch  cones. 

357.  The  operation  of  finding  the  tooth-outlines  upon  the  normal 
cone  may  be  performed  graphically,  as  in  Figs.  222  and  223.  Since 
the  tooth  projects  beyond  the  pitch  circle  to  R,  the  normal  cone 
must  be  correspondingly  extended  to  R,  which  determines  the  ex- 
treme diameter  of  the  blank  CTR ;  and  the  "bottoms  of  the  spaces 
upon  its  development  are  limited  by  the  circle  whose  radius  is  A'S', 
to  which  AS'  is,  of  course,  equal.  The  points  R,  P,  and  S,  in  re- 
volving about  AC}  describe  circles  which  in  the  side  view  appear  as 


APPROXIMATE   CONSTRUCTION   OF  TEETH. 


235 


the  right  lines  R T,  PF,  SCr ;  these  are  seen  in  their  true  form  and 
size  in  the  end  view,  Fig.  223,  which  it  is  necessary  to  construct  in 
order  to  draw  the  side  view.  Obviously,  the  length  of  the  arc  which 
measures  the  breadth  of  the  tooth  at  the  top,  at  the  bottom,  or  on 
the  pitch  circle,  will  be  the  same  in  the  end  view  as  in  the  develop- 
ment, the  chord  being  a  little  less  in  the  former.  The  same  holds 
true  in  regard  to  any  intermediate  circles  which  may  be  drawn,  and 
thus  the  outlines  of  the  teeth  in  the  end  view  may  be  determined  with 
any  required  degree  of  precision  ;  after  which  they  are  projected  to 
the  side  view,  the  tops  to  the  line  RT,  the  bottoms  to  the  line  SG, 
etc.,  in  the  usual  manner. 

Since  all  the  elements  of  the  tooth  run  to  the  vertex  0  of  the 


FIG.  223. 


pitch  cone,  it  follows  that  if  it  be  limited  at  the  smaller  end  by 
another  normal  cone,  as  in  the  figure,  its  outline  there  will  be  of  the 
same  form  as  that  of  the  outer  end  ;  and  the  mode  of  drawing  it  is 
sufficiently  indicated  without  further  explanation. 

358,  The  shaping  of  the  blank  by  the  addition  of  a  frustum  of  the 
normal  cone  TAR,  is  the  readiest  and  best  means  of  giving  a  present- 
able finish  to  the  teeth,  which  would  be  weak  at  the  extreme  points, 
and  project  beyond  each  other  in  a  very  unsightly  manner,  were  the 
larger  end  of  the  wheel  terminated  by  a  simple  transverse  plane. 

And  the  assumption  upon  which  this  method  of  laying  out  the 


23G 


EXACT   CONSTRUCTION   OF  TEETH. 


teeth  rests,  is  that  the  trace  upon  this  normal  cone,  of  the  surface 
generated  by  rolling  a  describing  cone  upon  the  pitch  cone,  will  upon 
development  become  a  true  epicycloid,  or  at  least  not  sensibly  differ 
from  it  within  the  limits  made  use  of  for  the  teeth.  Let  us  first, 
then,  see  how  the  exact  trace  of  this  surface  upon  the  normal  cone 
may  be  determined. 

359.  Construction  of  the  Correct  Tooth- Outline.— In  Fig.  224,  let  CPF 
be  the  pitch  cone,  GAH  the  normal  cone,  indefinitely  extended ; 
and  let  CL  be  the  axis  of  a  describing  cone,  ICK,  which  is 
tangent  to  the  pitch  cone  along  the  element  CP,  and  intersects 
the  normal  cone  in  the  curve  PON.  Making  a  projection  upon  a 
plane  perpendicular  to  CA9  the  circle  FP  is  seen  in  its  true  size  as 
F'TP,  the  curve  of  intersection  appearing  as  P'RN'S. 

Now  taking   CPI  as  the  describing  element,  it  is  clear  that  if  it 


FIG.  224. 

revolve  about  CL,  the  other  cones  remaining  fixed,  it  will  pierce  the 
normal  cone,  at  every  instant,  in  some  point  of  the  curve  PON. 
From  this  curve,  then,  the  required  trace  may  be  found  as  follows. 
Suppose  the  pitch  cone  fco  turn  as  shown  by  the  arrow,  through  an 
angle  measured  by  the  arc  P'a.  The  describing  element  CP  will 
thus  be  caused  to  revolve  about  CL,  through  an  angle  which,  the 
velocity  ratio  being  known,  can  be  readily  ascertained  ;  and  this  will 
enable  us  to  fix  the  point  #,  in  which  at  that  time  it  will  pierce  the 
normal  cone.  And  the  line  la  will  be  a  portion  of  the  required  trace. 
Let  the  rotation  progress  until  a,  reaches  the  position  c ;  then  since  la 
moves  with  the  pitch  cone,  it  will  at  that  instant  appear  as  ec  ;  and 
the  angular  motion  of  the  describing  element  meantime  being  known, 
we  can  find  the  point  h  upon  the  curve  P'RN',  in  which  it  will  then 


THE   RESULTS   COMPARED. 


237 


pierce  the  normal  cone,  and  extend  the  trace,  as  ced  :  which  operation, 
repeated  at  short  intervals,  will  enable  us  to  construct  the  true  form 
of  the  tooth-outline  with  any  stipulated  degree  of  accuracy,  within 
the  scope  of  graphic  operations. 

By  proceeding  in  like  manner  with  the  describing  cone  internally 
tangent  to  the  pitch  cone,  we  may  complete  the  tooth  by  finding  the 
trace  of  the  flank-surface. 

360.  Results  of  the  Two  Methods  Compared. — In  Fig.  225,  the  full 
outline  represents  the  tooth  of  a  wheel  laid  out  by  the  method  above 
explained  ;  and  the  dotted 
curves  are  the  forms  of  the 
faces  as  constructed  by  Tred- 
gold's  process. 

The  wheel  selected  for  this 
comparison  is  one  of  a  pair 
of  mitre-wheels,  as  those  are 
technically  called  which  are 
of  equal  diameters,  and  have 
their  axes  perpendicular  to 
each  other;  the  diameter  of 
the  base  of  the  pitch  cone 
being  30  inches,  the  number 
of  teeth  24. 

In  the  application  of  Tred- 
gold's  process,  we  are  at  lib- 
erty to  assume  a  describing 
circle  which  will  make  the 
flanks  of  the  developed  teeth 
radial,  its  diameter  being, 
therefore,  equal  to  the  slant 
height  of  the  normal  cone, 
and  this  was  done  in  laying  out  the  tooth  here  shown.  On  wrapping 
the  developed  sheet  back  upon  the  cone,  then,  these  radial  lines  will 
become  elements  of  that  surface ;  so  that  the  flanks  of  the  teeth  thus 
constructed  will  be  radial  planes. 

But  a  straight  line  can  be  generated  by  the  rolling  of  a  circle,  only 
when  it  rolls  within  another  circle  of  double  its  diameter,  in  its  own 
plane.  No  circular  cone,  then,  can  generate  a  perfect  plane  by  roll- 
ing within  another  one.  However,  if  the  angle  at  the  vertex  of  the 
describing  cone  be  half  that  at  the  vertex  of  the  pitch  cone,  the  sur- 
face swept  up  will  curve  so  slightly  near  the  first  element  of  contact, 
that  for  all  practical  purposes  the  flanks  may  be  regarded  as  truly 


FIG.  225. 


238  INSIDE   BEVEL   GEAR— INVOLUTE   TEETH. 

plane  surfaces.  But  now,  the  same  describing  cone  must  be  used  for 
the  face- surf  aces  ;  and,  as  shown  in  the  figure,  the  discrepancy  between 
the  results  of  the  two  methods  is  quite  decided,  and  sufficient  to 
affect  materially  the  constancy  of  the  velocity  ratio  if  the  approximate 
form  be  adopted. 

361.  Bevel  Wheels  in  Inside  Gear.— rln  laying  out  the  teeth  of  a  hol- 
low bevel  wheel,  by  Tredgold's  process,  it  is  apparent  that  the  size  of 
the  describing  circles  must  be  fixed  with  due  regard  to  the  limits 
deduced  for  the  case  of  annular  spur  gearing,  in  order  that  there  may 
be  no  interference  upon  the  development  of  the  normal  cone.     The 
spherical  epicycloids  and  hypocycloids,   it  is  true,  diifer  from  their 
plane  namesakes  in  this,  that  they  are  not  capable  of  two  generations ; 
so  that  the  peculiar  feature  of  double  tangency  between  the  face  and 
the  flank  of  a  pair  of  engaging  teeth  does  not  exist  in  inside  bevel 
gearing.     But  it  is  none  the  less  evident  that  if  the  outlines  of  the 
teeth  upon  the  developed  sheet  be  such  as  to  interfere  with  each  other, 
the  wheels  constructed  from  them  by  this  process  will  not  work. 

From  this  consideration  we  deduce,  as  a  safe  practical  rule  in  se- 
lecting the  describing  cones  v/hen  laying  out  the  teeth  by  the  exact 
method,  that  the  diameters  of  their  bases  should  not  be  greater  than 
those  of  the  describing  circles  which  can  be  employed  in  the  approxi- 
mate method  ;  the  circumferences  of  these  bases  being  tangent  to 
those  of  the  pitch  cones. 

A  pitch  cone  rolling  in  contact  with  a  plane  disk  (see  Fig.  81), 
presents,  upon  the  application  of  Tredgold's  process,  a  case  analogous 
to  that  of  a  rack  and  wheel  in  spur  gear,  and  requires  no  special 
notice.  In  the  use  of  the  exact  method,  the  details  of  the  operation 
are  substantially  the  same  as  already  explained ;  the  vertex  of  the 
cone  normal  to  the  plane  disk  being  infinitely  remote,  that  cone  be- 
comes a  cylinder,  but  no  additional  difficulties  result  from  the  intro- 
duction of  this  new  feature. 

362.  Bevel  Wheels  with  Involute  Teeth. — When  the  developed  bases 
of  the  normal  cones  are  taken  as  pitch  circles,  teeth  may  be  laid  out 
upon  them  by  the  involute  as  well  as  by  the  epicycloidal  system,  and 
the  wheels  thus  made  will  work  as  well  as  the  others.     The  surfaces 
of  their  teeth  are  obviously  approximations  to  those  which  would  be 
generated  by  the  rolling  of  a  plane  in  contact  with  two  base  cones. 
Such  a  rolling  plane  must,  clearly,  cut  the  plane  of  the  fixed  axes 
always  in  the  same  line,  and  this  line  will  bo  the  common  element  of 
two  pitch  cones  whose  diameters  will  have  the  same  ratio  as  those  of 
the  base  cones.     And,  taking  this  radius  of  the  rolling  plane  (whose 
motion  is,  during  the  rotation  of  the  cones,  one  of  revolution  about 


ACTION   OF   SPUR   AND   BEVEL  WHEELS   COMPARED.  239 

an  axis  perpendicular  to  the  plane,  passing  through  the  common 
vertex),  as  a  describing  line,  we  may  follow  its  movements,  and  by 
finding  the  points  in  which  at  given  intervals  it  pierces  the  normal 
cones,  determine  the  exact  tooth-outline  with  as  great  facility  and 
precision  as  in  the  preceding  case. 

This  tooth- surf  ace  may  be  also  generated  in  another  manner  ;  sup- 
pose the  pitch  cone  to  be  cut  from  vertex  to  base  along  an  element, 
and  the  surface  to  be  unrolled  into  a  plane  ;  this  plane  will,  during 
the  process,  be  always  tangent  to  the  cone,  the  c#t  edge  will  sweep 
up  the  surface  under  consideration,  and,  since  it  is  of  constant  length, 
its  extremity  will  trace  a  curve  which  may  be  properly  called  a  spher- 
ical involute. 

It  is  not  necessary  to  go  into  a  detailed  examination  of  the  form 
of  the  surface  which  by  rolling  in  contact  with  the  pitch  cones  would 
generate  teeth  of  the  forms  here  spoken  of,  since  the  method  above 
given  is  much  more  simple.  The  close  analogy  between  the  plane 
and  the  spherical  involute,  however,  is  sufficient  to  make  it  apparent 
that  such  a  conical  surface,  whose  spherical  directrix  will  have  a 
corresponding  analogy  to  the  logarithmic  spiral,  is  capable  of  thus  de- 
scribing these  identical  teeth,  which  consequently  form  no  exception 
to  the  general  law. 

363.  Action  of  Bevel  and  Spur  Wheels  Compared. — It  is  true  of  bevel 
as  well  as  of  spur  gearing,  that  the  smaller  and  more  numerous  the 
teeth,  the  better  will  be  the  action,  because  there  will  be  less  of  both 
sliding  and  obliquity. 

Now  experience  shows  that,  other  things  being  equal,  a  pair  of 
bevel  wheels  will  run  more  smoothly,  and  in  a  general  way  operate 
more  satisfactorily,  than  a  pair  of  spur  wheels  of  the  same  numbers 
of  teeth.  The  reason  of  this  will  be  seen  very  clearly  by  reference  to 
Fig.  221,  if  we  imagine  the  outer  ends  of  the  teeth  to  be  bounded  by 
the  surface  of  the  sphere,  and  confine  our  attention  to  the  action 
upon  each  other  of  the  outlines  thus  formed.  The  point  of  contact, 
throughout  the  engagement  of  any  one  pair  of  teeth,  will  always  lie 
in  or  near  to  the  plane  AB9  perpendicular  to  CP,  upon  which  are 
developed  the  normal  cones,  which  latter  in  their  turn  do  not,  within 
the  limits  of  the  height  of  the  teeth,  deviate  greatly  from  the  spheri- 
cal surface.  The  result  of  this  is,  that  although  the  velocity  ratio  is 
determined  by  the  lengths  of  the  perpendiculars  PG,  PH,  the  dura- 
tion of  the  action  is  very  nearly  the  same  as  it  would  be  were  the 
developed  teeth  to  work  together  in  their  own  plane  about  the  centres 
A',  B'.  The  action  of  these  wheels,  then,  in  so  far  as  it  is  affected 
by  the  numbers  of  the  teeth,  is  substantially  the  same  as  that  of  spur 


240  COKICAL  LOBED   WHEELS. 

wheels  of  the  same  pitch,  with  PA,  PB,  for  the  radii  of  the  pitch 
circles.  Or,  in  general,  the  action  of  a  bevel  wheel  of  a  given  num- 
ber of  teeth,  is  in  this  respect  equivalent  to  that  of  a  spur  wheel  hav- 
ing a  number  greater  in  the  proportion  of  the  slant  height  cf  the 
normal  cone  to  the  radius  of  its  base,  or  of  the  slant  height  of  the 
pitch  cone  to  its  altitude. 

364.  Teeth  of  Conical  Lobed  Wheels. — If  a  cone  of  any  form  be  inter- 
sected by  a  sphere  whose  centre  is  at  the  vertex,  every  point  of  the  in- 
tersection is  equally  distant  from  that  centre  ;  let  a  describing  cone  be 
rolled  upon  the  first  one,  as  in  Fig.  220,  then  the  extremity  of  any  one 
of  the  elements  of  the  rolling  cone  will  trace  upon  the  sphere  a  curve, 
which  may  be  considered  as  the  directrix  of  the  conical  surface  genera- 
ted by  the  element  itself ;  and  that  surface  will,  in  accordance  with 
the  general  principle,  be  of  the  form  required  for  the  tooth,  the  first 
or  base  cone  being  regarded  as  a  pitch  surface  ;  and,  without  going 
into  details,  it  is  apparent  that  the  determination  of  the  spherical 
directrices  may  be  effected  by  graphic  means. 

Otherwise ;  let  a  plane  be  passed,  say  perpendicular  to  the  fixed 
axis  of  rotation,  or  to  the  line  of  symmetry,  if  there  be  one,  thus 
forming  a  base  for  the  pitch  cone.  Inasmuch  as  its  contour  may  be 
practically  made  up  of  approximating  circular  arcs,  the  pitch  cone 
may  be  considered  as  made  up  of  portions  of  various  circular  cones  ; 
and  these  may  be  treated  as  previously  explained,  by  either  the  exact 
or  the  approximate  method.  The  normal  cones,  upon  which  the 
traces  of  the  teeth  are  thus  obtained,  will  not  have  a  common  vertex  ; 
but  since  each  is  to  be  dealt  with  separately,  this  is  of  no  consequence. 

There  is  no  instance  to  be  cited  of  the  actual  employment  of  these 
non-circular  conical  wheels ;  which  indeed  have  not  been  described 
by  any  previous  writer.  An  objection  might  be  urged  on  account  of 
the  difficulty  of  making  them  :  which,  however,  does  not  lessen  the 
abstract  interest  attaching  to  them  as  a  new  class  of  kinematic  com- 
binations. The  above  suffices  to  show  that  the  graphic  operations 
relating  to  them  are  sufficiently  simple  in  detail,  although  tedious  by 
reason  of  their  number ;  and  it  may  be  questioned  whether  they 
would  really  prove  more  difficult  to  make  than  the  skew  wheels 
which  are  occasionally  met  with. 

365.  Methods  of  Cutting  the  Teeth. — Since  the  teeth  taper  from  end 
to  end,  the  outlines  of  the  transverse  sections  are  continually  chang- 
ing ;  it  is  therefore,  impossible  to  cut  them  with  perfect  accuracy  by 
means  of  a  milling  cutter  of  the  usual  form.     Nevertheless,  this  is 
the  method  almost  universally  adopted  when  they  are  cut  at  all.     In 
order  to  distribute  the  unavoidable  errors  as  uniformly  as  may  be,  it 


METHODS  OF  CUTTING  BEVEL  TEETH.  241 

is  customary  to  make  the  outline  of  the  cutter  agree  in  form  with  the 
cross  section  of  the  tooth  at  the  middle  of  its  length  ;  its  travel  is 
directed  along  the  line  joining  the  face  and  flank  of  the  tooth,  that 
is  to  say  along  the  element  of  the  pitch  cone,  first  on  one  side  of  the 
space,  and  then  on  the  other,  the  thickness  of  the  cutter  being  less 
than  the  breadth  of  the  space. 

The  effect  of  this  is,  that  at  the  larger  end  of  the  tooth  the  flank 
is  too  full  and  the  face  not  full  enough,  while  at  the  smaller  end  these 
errors  are  reversed  ;  besides,  the  tooth  surfaces  thus  cut  are  not  conical, 
but  cylindrical,  all  the  elements  being  parallel  to  the  line  along  which 
the  travel  of  the  cutter  is  directed.  These  surfaces,  consequently, 
will  not  work  in  true  line  contact,  except  at  the  one  instant  when  the 
junctions  of  faces  and  flanks  reach  the  plane  of  the  axes,  until  they 
have  "worn  to  a  bearing"  by  use  ;  but  neither  then  nor  thereafter 
will  they  work  correctly. 

366.  Corliss's  Bevel  Gear  Cutting  Engine. — There  are,  however,  ways 
and  means  of  accomplishing  better  results  than  this;  and  to  the  in- 
genuity and  enterprise  of  Mr.  George  H.  Corliss,  of  Providence,  R.  L, 
is  due  the  production  of  an   engine,    capable  of  doing  the  work 
with  absolute  theoretical  precision.     In  this  admirable  machine  the 
milling  cutter  plays  no  part,  but  the  teeth  are  planed  out,  element 
by  element. 

Without  going  into  details,  it  will  be  understood,  first,  that  the 
cutting  point  of  the  tool  is  made  to  travel  always  in  a  right  line  pass- 
ing through  the  vertex  of  the  pitch  cone  ;  it  must,  therefore,  at  each 
cut  plane  out  an  element  of  a  conical  surface.  Second,  that  the 
motions  are  so  controlled  with  reference  to  a  guide  template,  that  a 
line  drawn  from  the  vertex  through  the  cutting  point  shall  always 
touch  the  outline  of  the  template,  which  is,  then,  the  directrix  of 
this  conical  surface. 

Now,  all  that  is  necessary  to  the  production  of  a  perfect  wheel,  is 
to  provide  this  machine  with  a  template  whose  outline  accurately 
conforms  to  that  of  a  transverse  section  of  a  tooth.  This,  it  should 
also  be  stated,  is  practically  laid  out  upon  a  larger  scale  than  if  the 
outer  end  of  the  tooth  were  to  be  drawn  in  the  usual  manner,  because 
it  is  placed  in  the  engine  considerably  farther  than  the  base  of  the 
blank  from  the  vertex  of  the  pitch  cone  ;  thus  reducing  the  propor- 
tional magnitude  of  any  unavoidable  errors  in  the  graphic  processes. 

367.  Twisted  Bevel  Wheels.— The  feasibility  of  placing  upon  the 
same  shafts  a  series  of  bevel  wheels  cut  from  the  same  cones,  but  in 
different  phases  of  their  action,  is  self-evident.     The  advantages  of 
such  stepped  wheels  would  be  exactly  the  same  as  in  the  similar  ar- 

16 


242 


TWISTED   BEVEL  WHEELS. 


rangement  of  spur  gearing  ;  but  there  are  practical  drawbacks  which, 
it  may  be  admitted,  would  prevent  their  employment.  But  the  pro- 
cess of  twisting,  by  which  the  spur  wheels  are  transformed  into 
Hooke's  spiral  wheels,  may  be  equally  well  applied  to  conical  ones  ; 
and  to  the  form  of  bevel  gearing  thus  produced  no  reasonable  objection 
can  be  made,  except  the  difficulty  of  making  it ;  and  this,  as  will 
presently  appear,  is  not  insuperable,  nor  so  great  as  might  be  supposed. 
In  Fig.  226,  it  is  evident  that  if  each  plate  of  the  vertical  pitch 
cone  be  twisted  through  the  same  angle,  it  will  drive  the  correspond- 
ing plate  of  the  inclined  cone  through  an  angle  depending  upon  the 
relative  diameters  ;  so  that  if  these  plates  be  indefinitely  increased  in 

number,  the  rectilinear  elements  AB, 
EF9  will  become  conical  helices,  as 
shown. 

The  graphical  construction  pre- 
sents no  difficulty ;  we  have  in  each 
case  a  uniform  rotation,  combined 
with  a  uniform  advance  along  an 
clement.  Consequently  since  if  the 
amount  of  twist  upon  one  cone  be 
assumed  that  on  the  other  is  known, 
both  curves  are  readily  drawn  ;  they 
move  in  perfect  rolling  contact,  and 
develcpe  upon  the  common  tangent 
plane  into  the  same  Archimedean 
spiral. 

Just  as  with  the  spur  wheels,  any 
form  of  tooth  which  will  work  cor- 
rectly before  twisting  will  do  so  after- 
ward ;  and  if  the  faces  be  purposely 
made  not  full  enough  to  be  conjugate  to  the  engaging  flanks,  there 
will  be  at  each  instant  but  a  single  point  of  tangency,  which  will 
alway  lie  in  the  plane  of  the  axes,  and  the  wheels  will  run  in  pure 
rolling  contact. 

Now,  although  such  teeth  as  these  cannot  be  made  by  the  use  of  a 
milling  cutter,  it  will  be  apparent  that  if  the  blank  be  placed  in  the 
Corliss  machine  above  described,  and  given  a  motion  of  uniform  rota- 
tion during  the  advance  of  the  cutting  point,  they  can  be  planed  out 
as  easily  and  as  accurately  as  any  others. 

The  Teeth  of  Skew   Wheels. 
368,  The  Problem  of  Determining  the  Forms  of  Teeth  which  shall  work 


FIG.  226. 


TEETH    OF   SKEW   WHEELS — ANALOGICAL   THEOKY.  243 

in  right-line  contact  while  transmitting  rotation  about  fixed  axes,  pre- 
sents itself  in  a  new  form  when  those  axes  lie  in  different  planes. 

All  wheels  with  teeth  which  act  in  this  manner  must  ultimately 
reduce  to  pitch  surfaces  whose  class  depends  upon  the  relative  posi- 
tions of  the  axes ;  these  will  be  tangent  to  each  other  along  a  right 
line,  and  move  in  contact  of  which  the  sliding,  if  there  be  any,  is  also 
along  the  common  element.  A  third  surface  of  the  same  class  being 
placed  in  contact  with  these  along  the  same  line,  will  move  in  contact 
of  the  same  nature  with  either  or  both  ;  and  if  one  pitch  surface 
drive  the  other,  the  motion  of  the  third  is  the  same  whether  we 
regard  it  as  derived  from  the  first  or  the  second.  When  the  velocity 
ratio  is  constant,  these  will  be  surfaces  of  revolution — cylinders,  cones, 
or  hyperboloids ;  and  in  either  case  all  three  rotate  in  contact  about 
fixed  axes. 

During  such  rotation,  when  the  axes  are  parallel,  an  element  of  the 
third  cylinder,  by  its  motion  relatively  to  the  pitch  cylinders,  simul- 
taneously generates  the  conjugate  surfaces  for  the  teeth  of  spur 
wheels.  Now  the  cylinder  is  the  limiting  form  assumed  by  the  cone 
when  the  vertex  is  infinitely  distant ;  and  accordingly,  when  the  axes 
intersect,  the  third  cone  in  like  manner  describes  the  surfaces  for  the 
teeth  of  bevel  wheels. 

Again,  the  cone  is  but  the  special  case  of  the  hyperboloid  in  which 
the  generatrix  approaches  infinitely  near  to  the  axis,  and  by  pursuing 
the  analogy  we  should  reach  the  deduction  that  an  element  of  the 
third  hyperboloid,  if  taken  as  a  describing  line,  will  generate  the  con- 
jugate tooth-surfaces  for  skew  wheels. 

369.  This  conclusion  is  plausible,  and  has  received  the  indorsement 
of  high  authorities.  Thus,  Prof.  Willis  says  :  * 

"  The  surfaces  adapted  for  teeth  in  the  case  of  rolling  hyperboloids 
might  be  obtained  in  a  manner  similar  to  those  of  rolling  cones,  by 
taking  an  intermediate  describing  hyperboloid  ;  but  it  does  not  appear 
possible  to  derive  from  this  any  rules  sufficiently  simple  for  appli- 
cation." 

He  adds  that  a  sufficiently  close  approximation  may  be  made  by 
drawing  two  cones  normal  to  the  hyperboloidal  frustum  selected,  de- 
veloping these,  and  after  laying  out  teeth,  as  in  Tredgold's  method  for 
bevel  wheels,  wrapping  them  back  in  their  proper  relative  positions. 
The  forms  and  proportions  of  the  teeth  traced  upon  the  larger  and 
the  smaller  cone  respectively,  are  presumably  the  same  ;  and  the  tooth- 
surface  is  composed  of  right  lines  joining  the  corresponding  points  of 

*  Principles  of  Mechanism,  p.  151. 


244 


TEETH   OF   SKEW   WHEELS. 


their  contours  after  the  developments  are  restored  to  their  original 
conical  forms,  in  due  relation  to  each  other  as  determined  by  the  gen- 
eratrix of  the  pitch  hy- 
perboloid. 

Prof.  Kankine,*  again, 
remarks:  "  The  surf  aces 
of  the  teeth  of  a  skew- 
bevel  wheel  belong,  like 
its  pitch  surface,  to  the 
hyperboloidal  class,  and 
may  be  conceived  to  be 
generated  by  the  motion 
of  a  straight  line  which, 
in  each  of  its  successive 
positions,  coincides  with 
the  line  of  contact  of  a 
tooth  with  the  corre- 
sponding tooth  of  an- 
other wheel.  Those 
surfaces  may  also  be  con- 
ceived to  be  traced  by 
the  rolling  of  a  hyper- 
boloidal roller  upon  the 
hyperboloidal  pitch 
surface." 

And  he  in  turn  pro- 
ceeds to  describe  at 
some  length  a  process 
of  constructing  the  teeth 
which,  although  differ- 
ing in  detail  from  that 
above  explained,  gives 
nearly,  and  in  some 
cases  precisely,  the  same 
results  ;  both  are,  then, 
to  be  regarded  as  ap- 
proximations more  or 
less  close. 

370.  Direct  Process  of 
Construction. — But,    as- 
suming that  the  surfaces  generated  by  the  use  of  an  auxiliary  hyper- 


Fic.  227. 


Machinery  and  Mill- Work,  p.  146. 


DIRECT  PROCESS  OF  CONSTRUCTION. 


245 


boloid  are  correct,  it  is  nearly,  if  not  quite,  as  easy  to  construct  them 
directly  as  in  the  exact  method  for  bevel  wheels  ;  supposing  the  pitch 
surface  and  the  generating  surface  to  rotate  in  contact,  the  axes  being 
fixed,  we  have  simply  to  follow  the  movements  of  the  describing  line, 
and  find  the  points  in  which  it  pierces  the  normal  cones  at  various 
phases  of  the  action. 

In  Fig.  227,  V,  Y  are  the  vertices  of  the  cones  normal  to  the  verti- 
cal pitch  surface,  and  the  inclined  describing  hyperboloid,  touching 
the  other  internally,  is  shown  in  position  for  generating  the  flank  of 
the  tooth. 

For  convenience,  the  velocity  ratio  of  these  two  hyperboloids  is 
taken  as  2  to  1 ;  hence  a  rotation  of  the  vertical  surface  through  the 
angle  BVP,  will  cause  the  other  to  turn  through  the  angle  BCO 
twice  as  great.  The  describing  line,  which  was  originally  at  AS,  will 
then  have  the  position  720,  piercing  the  upper  normal  cone  in  some 
point  $,  which  being  found,  occupies  a  known  position  in  relation  to 
the  element  VP  ;  and  a  point  similarly  situated  with  reference  to  the 
element  VB,  will  evidently  lie  in  the  trace  of  the  required  flank  upon 
that  cone.  By  repeating  this  process  any  desired  number  of  points  in 
the  curve  may  be  located  ;  and  applying  it  to  the  lower  normal  cone, 
we  determine  the  outline  of  the  smaller  end  of  the  tooth.  Or  by  ex- 
tending the  describing  line,  and  finding  where  it  pierces  the  plane 
of  the  gorge  circle,  we  may  in  a  similar  manner  map  out  the  trace 
upon  that  plane. 

371.  Next,  the  describing  surface  being  placed  in  external  contact 
along  the  same  line,  AB,  and  the  normal 
cone  being  sufficiently  extended,  the  face 
of  the  tooth  is  constructed  in  a  similar 
manner,  needing  no  illustration.  The 
method  of  finding  the  point  in  which  the 
describing  line  pierces  the  normal  cone  is 
shown  in  Fig.  228.  V  being  the  vertex  of 
the  cone,  AB  its  base,  and  MN  the  line, 
draw  through  Fa  line,  VO,  either  parallel 
to  or  intersecting  MN;  this  line  pierces  the 
plane  of  the  cone's  base  in  0,  and  MN 
pierces  it  in  P.  Therefore  the  plane  deter- 
mined by  MN  and  V.O  cuts  the  plane  AB 
in  the  line  OP,  which,  produced  if  neces- 
sary, cuts  the  circumference  of  the  base  in 
the  point  L  ;  now  drawing  VL,  that  element 
of  the  cone  cuts  the  given  line  MN'm  S,  which  is  the  required  point. 


M 


FIG.  228. 


TEETH   NOT   SYMMETRICAL. 


372.  The  Teeth  not  Symmetrical. — It  will  be  perceived  that  the  trace 
of  the  tooth  upon  the  normal  cone  is  derived  from  the  intersection  of 
that  cone  with  the  describing  surface,  just  as  it  was  in  constructing 
the  correct  teeth  for  bevel  wheels.     In  that  case  the  axes  of  these  two 
surfaces  were  in  the  same  plane,  which  divided  the  curve  of  intersec- 
tion symmetrically.     But  this  is  not  so  with  the  hyperboloid,  and  in 
Fig.  227  that  curve  will  not  be  symmetrical  with  reference  to  the  ele- 
ment VB  of  the  normal  cone.     The  result  of  this  is,  that  the  traces  of 
the  two  flank  surfaces,  derived  from  the  parts  of  the  curve  on  opposite 
sides  of  the  point  B,  will  not  be  similar.     The  same  holds  true  in 
regard  to  the  face  surfaces,  so  that  the  trace  of  the  complete  tooth  upon 
the  normal  cone  will  not  be  symmetrical  to  a  radius,  as  it  is  in  the 
constructions  of  Prof.  Willis  and  Prof.  Eankine. 

This  may  be  seen  from  another  point  of  view,  thus  :  The  inclina- 
tions of  the  describing  line  to  the  elements  of  the  normal  cone  will 
not.  vary  in  the  same  manner  nor  in  the  same  degree  in  the  opposite 
directions  of  its  motion  from  the  position  AB.  The  difference  of  form 
is  not  great  for  the  flanks,  but  is  more  conspicuous  in  the  faces,  as 
might  be  expected  from  their  greater  length  and  more  rapid  changes 
of  curvature. 

373.  Determination  of  Height  of  Tooth.— In  Fig.    229,    let  D  be 
the  axis  of  a  wheel,   perpendicular  to  the  paper.     LBO  the  base 

of  the  normal  cone, 
AB  the  generatrix 
of  the  pitch  surface, 
BT  the  intersection 
of  the  normal  cone 
with  the  describing 
hyperboloid,  from 
which  is  derived  the 
trace  BFG  of  the  face 
13  of  the  tooth.  The  ac- 
tion of  this  face  begins 
at  B,  and  BT  is  the  locus  of  contact ;  hence  if  the  angle  of  recess, 
BDP,  be  given,  we  draw  through  P  a  curve  similar  to  BG,  cutting 
BT\i\  E,  which  will  be  the  highest  point  of  the  face.  Conversely, 
if  the  height  BF  of  the  tooth  be  assumed,  an  arc  through  F  about 
the  centre  D  cuts  BT  in  E,  and  FDE  or  its  equal  BDP  will  be  the 
angle  of  action  between  this  face  and  the  engaging  flank  of  the  other 
wheel. 

Let  C  be  the  axis  of  that  second  wheel,  also  perpendicular  to  the 
paper  ;  HK  the  generatrix  (coincident  with  AB  when  the  wheels  are 


FIG.  229. 


HEIGHT  OF  TOOTH   AtfD   FORM   OF   BLANK.  247 

in  position) ;  HU  the  trace,  on  the  second  normal  cone,  of  the 
same  describing  hyperboloid  that  was  used  for  the  face  of  the  first 
wheel :  and  HN  t\\Q  trace  of  the  flank.  Now  the  turning  of  the  first 
wheel  through  the  given  angle,  BDP,  causes  the  describing  surface  to 
turn  through  a  known  angle  ;  which  enables  us  to  find  the  point  J  in 
HU,  corresponding  to  that  angular  motion.  Then  an  arc  about 
centre  C  through  this  point  /,  cuts  HN  in  R,  which  limits  the  acting 
depth  of  the  flank.  Or,  otherwise  ;  the  given  rotation  of  the  first 
wheel  will  drive  the  second  through  a  known  angle  HCS ;  then  a  curve 
through  S,  similar  to  HN,  cuts  ITU  in.  /;  and  thus  the  acting  depth 
is  found  without  reference  to  the  rotation  of  the  describing  surface. 

The  same  process  is  to  be  repeated  with  respect  to  the  face  of  the 
second  wheel  and  the  flank  of  the  first,  which  completes  this  op- 
eration and  determines  the  whole  angle  of  action  for  each  wheel. 
But  the  flank-curves  are  to  be  continued  beyond  the  points  thus 
found,  because,  as  in  any  other  form  of  gearing,  clearing  spaces  must 
be  provided  in  each  wheel,  for  the  passage  of  the  teeth  of  the  other. 

374,  Determination  of  the  Form  of  the  Blank. — When  the  height  of 
the  tooth  has  thus  been  ascertained,  or  assumed,  the  position  of  the 
describing  line  which  passes  through  the  outer  point  of  its  face  is  defi- 
nitely fixed.     Now,  this  element  of  the  tooth-surface,  in  revolving 
around  the  fixed  axis  of  the  wheel,  generates  a  new  hyperboloid,  and 
the  blank  should  be  made  in  the  form  of  a  frustum  of  that  surface, 
limited  by  the  cones  normal  to  the  pitch  hyperboloid.     For  otherwise 
the  teeth  will  begin  and  end  their  action  at  a  single  point,  instead  of 
along  a  line  :  the  obvious  result  being  that  the  points,  especially  of 
the  follower's  teeth,  will  suffer  from  abrasion. 

If  the  wheels  are  located  in  the  immediate  neighborhood  of  the 
gorge  planes,  the  true  hyperbolic  outline  must  be  carefully  followed 
in  making  the  blank,  in  order  to  avoid  this  danger.  But  if  not,  the 
hyperbola  becomes  so  flat  at  a  short  distance  from  the  vertex,  that  it 
will  ordinarily  suffice  in  practice  to  use  a  frustum  of  a  cone  tangent 
to  this  external  hyperboloid,  at  the  middle  of  the  length  of  the  tooth 
as  measured  along  its  outer  element. 

375.  This  is  illustrated  in  Fig.  230 ;  the  dotted  curve  EF  is  the 
outline  of  the  pitch  surface,  to  which  the  cones  whose  vertices  are 
V  and  Y  are  normal,  and  ON  is  the  outline  of  the  external  hyper- 
boloid,  generated   by  revolving    the   describing  line  which  passes 
through  N  the  highest  point  of  the  tooth,  about  the  vertical  axis  VX. 
NM  being  the  length  of  the  outer  hyperbola  to  be  used,  let  this  be 
bisected  at  R,  at  which  point  draw  a  tangent  to  the  curve  ;  this  will 
intersect  the  axis  at  X,  the  vertex  of  the  tangent  cone. 


248 


FORMS   OF  THE  COMPLETE  TEETH. 


The  method  of  drawing  the  tangent  to  the  hyperbola  is  also  shown. 
Since  the  position  of  the  generatrix  of  the  outer  surface  with  respect 
to  the  axis  is  known  from  the  antecedent  constructions,  it  will,  when 
parallel  to  the  paper  as  in  the  diagram,  take  a  known  direction,  AB, 
asymptotic  to  the  hyperbola  ON.  The  companion  generatrix  then 
becomes  the  other  asymptote  AL  ;  now  draw  through  R  a  parallel  to 
AB,  cutting  AL  in  P,  and  on  AL  set  off  PC  =  AP,  then  RC  is  the 

required  tangent. 

376.  The  forms  of  the 
teeth  for  a  pair  of  wheels, 
laid  out  by  the  processes 
above  described,  are  shown 
in  Figs.  231  and  232,  the 
wheels  being  seen  from 
the  smaller  end;  the  long 
tooth  011  the  right,  in  the 
upper  part  of  each  figure, 
is  continued  till  cut  by  the 
gorge  plane,  the  shorter 
teeth  on  the  left  being  lim- 
ited by  an  inner  normal  cone 
as  in  Fig.  230.  In  the 
lower  part  of  each  figure 
are  shown  the  common  ele- 
ment of  the  pitch  surfaces, 
the  trace  of  one  side  of  the 
tooth  upon  the  outer  normal 
cone,  and  the  parts  of  the 

FlG.  030.  curves  of  intersection  used 

in  the  construction.  The  lines  employed  in  the  determination  of  the 
other  side  of  the  tooth  are  omitted  in  order  to  prevent  confusion ; 
but  it  is  proper  to  state  that  these  figures  are  copied  from  drawings 
executed  with  great  care  upon  a  large  scale,  in  exact  accordance  with 
the  methods  above  set  forth.  The  conditions  assumed  are,  that  the 
projections  of  the  axes  upon  a  plane  parallel  to  both,  shall  cross  each 
other  at  an  angle  of  60°,  as  in  Fig.  97,  and  that  the  wheels  shall  have 
eighteen  and  twenty-seven  teeth  respectively. 

It  is  to  be  noted  that  in  consequence  of  the  peculiar  relation 
between  the  pitch  and  describing  surfaces,  not  only  are  the  fronts  and 
backs  unlike,  but  the  traces  on  the  various  normal  cones  are  also  dis- 
similar, the  whole  form  of  the  tooth  changing  with  the  distance  from 
the  gorge  plane. 


THE   ANALOGICAL   THEORY   WRONG.  249 

377.  Fallacy  of  the  Preceding  Construction. — The  correctness  of  the 
above  method  of  generating  the  conjugate  tooth-surfaces  for  skew 


FIG.  231. 

wheels  has  hitherto  passed  unchallenged ;  a  fact  perhaps  not  to  be 

wondered  at,  so  plausible 

is  the  analogical  reasoning 

by  which  the  deduction  is 

reached. 

Either  one  is  unquestion- 
ably correct  in  relation  to 
the  describing  hyperbo- 
loid.  Let  the  upper  and 
lower  bases  of  the  latter  be 
two  discs  fixed  upon  one 
axis,  and  in  these  let  a 
single  wire  be  secured,  to 
represent  the  describing 
line.  If  this  axis  be  prop- 
erly placed  in  relation  to 
either  pitch  surface,  it  will  FIG.  232. 

be  correctly  driven  by  the  corresponding  tooth-surface  acting  against 


250 


ROLLING   OF   OtfE   IIYPERBOLOID   UPOK   ANOTHER. 


the  wire,  after  the  manner  of  a  pin-wheel  in  spur  gearing.  And  by 
hypothesis  the  engaging  surfaces  have  at  every  instant  in  common 
this  describing  line,  by  which  they  were  simultaneously  swept  up 
during  the  action. 

So  far,  well ;  but  one  thing  more  is  necessary — this  common  line 
must  be  a  line  of  tangency  ;  and  it  appears  to  have  been  assumed  that 
this  is  the  case,  probably  because  up  to  this  point  the  analogy  is  perfect. 
Every  link  in  the  chain  has  stood  the  test ;  now  let  us  examine  the  hook. 
378.  It  is  necessary,  first,  to  gain  a  clear  idea  of  what  occurs  when 
one  hyperboloid  rolls  around  another  which  is  stationary  ;  a  question 

not  previously  considered. 

If  the  two  surfaces  shown 
in  Fig.  233  turn  as  shown  by 
the  arrows  (the  axes  being 
for  the  moment  regarded  as 
fixed),  let  AL,  A I  be  the 
linear  motions  of  the  points 
which  fall  together  at  A  ; 
these  will  have  the  same  com- 
ponent, AKy  perpendicular  to 
the  common  element,  AB, 
and  IL  represents  the  amount 
of  sliding  between  the  two 
hyperboloid s  along  that  ele- 
ment. 

Now  suppose  the  motion 
of  the  inclined  surface  to  be 
suddenly  arrested,  and  the 
axis  of  the  vertical  one  to  be 
at  the  same  instant  released 
from  its  bearings,  its  rotation 
continuing.  The  latter  must 
then  revolve  around  the  axis 
of  the  former  in  the  direction 
opposite  to  that  in  which  the 
former  was  turning,  and  with 
the  same  angular  velocity. 
The  point  A  of  the  vertical  hyperboloid  will  then  have  a  component 
motion,  AL,  due  to  its  original  rotation,  and  another,  AH,  equal  and 
opposite  to  AT,  due  to  the  revolution. 

The  resultant  of  these  is  A  0,  in  BA  produced,  and  equal  to  IL. 
This  is  as  it  should  be,  evidently,  since  the  relative  sliding  of  the  two 


FIG.  233. 


THE   ANALOGICAL  THEORY  WRONG. 

surfaces  must  remain  the  same ;  but,  moreover,  the  motion  of  every 
point  in  or  connected  with  the  travelling  hyperboloid  must  have  a 
component  equal  and  parallel  to  AO.  The  only  other  motion  of 
which  any  such  point  is  capable  is  one  of  revolution  about  AB,  and 
since  the  angular  velocity  must  be  the  same  for  every  point,  it  will 
suffice  to  find  it  for  one. 

The  point  E  has  a  component  motion,  EP,  equal  and  opposite  to 
AL,  due  to  the  rotation  ;  and  another  due  to  the  revolution,  in  the 
direction  of  AH,  but  greater,  in  the  proportion  of  DE  to  DA.  The 
resultant  is  DT,  the  tangent  to  the  helix  which  the  point  E  is  at  the 
instant  describing.  Therefore  OT,  perpendicular  to  AB,  is  the  linear 
velocity  of  the  rotation  of  this  point  about  the  common  element  of  the 

OT 

two  surfaces,  and  -r-=  is  the  angular  velocity  for  every  point  in  the 
AJl/ 

rolling  hyperboloid. 

379.  Precisely  the  same  result  has  already  been  reached  in  a  differ- 
ent manner,  as  will  be  seen  by  reference  to  Fig.  25,  and  the  accompa- 
nying text.     The  nature  of  the  action  is  most  explicitly  stated  by 
Prof.  Rankine,  thus  *  :  *'  If  one  of  those  bodies  is  fixed,  and  the  other 
made  to  roll  upon  it,  they  continue  to  touch  each  other  in  a  straight 
line,  which  is  the  instantaneous  axis  of  the  rolling  body,  and  the  rota- 
tion about  that  instantaneous  axis  is  accompanied  by  a  sliding  motion 
along  the  same  axis  so  as  to  give,  as  the  resultant  compound  motion, 
a  helical  motion  about  the  instantaneous  axis." 

If,  then,  we  take  any  element  of  the  rolling  hyperboloid  as  a  describ- 
ing line,  it  is  at  any  instant  in  the  act  of  generating  a  helicoid,  which 
will  be  tangent  all  along  this  line  to  the  tooth-surface  generated  by  a 
continuation  of  the  rolling.  Therefore  the  plane  tangent  to  the  latter 
surface  at  any  point,  is  determined  by  the  describing  line  itself,  and 
the  tangent  to  the  helix  at  that  point  constructed  as  above. 

380.  Now  in  generating  the  conjugate  tooth-surfaces  for  a  pair  of 
wheels,  the  describing  hyperboloid  rolls  upon  the  outside  of  one  pitch 
surface,  and  upon  the  inside  of  the  other,  as  in  Fig.  234.     The  direc- 
tion of  the  rotation  about  the  instantaneous  axis  is  of  course  the  same 
in  both  cases,  but  that  of  the  sliding  is  not ;  in  the  rolling  upon  the 
inclined  pitch  surface  the  sliding  is  represented  by  AO,  while  in  the 
rolling  within  the  vertical  surface  it  is  represented  by  AP,  in  the  op- 
posite direction. 

In  Fig.  235  let  the  vertical  line  AB  represent  the  common  element 
of  the  pitch  surfaces  or  instantaneous  axis,  and  OE  the  describing 

*  Machinery  and  Millwork,  p.  71. 


252 


THE   CONJUGATE   SURFACES   INTERSECT. 


line,  the  rotation  being  indicated  by  the  arrows.  The  tangent  to  the 
helix  described  at  the  instant  by  the  point  P  will  lie  in  the  plane  tan- 
gent to  the  cylinder  upon  which  that  helix  lies,  and  in  the  horizontal 
projection  will  have  the  direction  P'l'  perpendicular  to  P'A'.  Let 
the  motion  parallel  to  AB  be  upward,  then  PI  will  be  the  vertical 
projection  of  this  tangent.  Let  MN  be  a  horizontal  plane  ;  it  cuts 
PI  at  L,  and  OE  at  0,  giving  O'L'  as  the  trace  upon  MN  of  the 


M 


Fro.  234. 


FIG.  235. 


plane  tangent  to  the  helicoid  at  P.  If,  on  the  other  hand,  P  have  a 
downward  motion  parallel  to  AB,  PI'  will  be  horizontal  and  P//the 
vertical  projection  of  the  tangent  to  the  helix,  which  pierces  MN  at 
R,  giving  O'R  as  the  trace  of  this  tangent  plane,  which  cuts  the  first 
one  in  the  line  OE.  A  vertical  plane  through  the  moving  line  will 
evidently  also  contain  the  tangents  to  the  helices  described  by  the 
point  E  of  that  line,  which  is  at  the  least  distance  from  AB.  At  this 
point,  then,  and  at  no  other,  will  the  two  helicoids  have  a  common 
tangent  plane. 

381.  Consequently  the  hook  breaks,  the  chain  is  useless,  and  these 
teeth  will  not  work  at  all.  The  common  line  of  the  engaging  face 
and  flank  is  a  line  not  of  tangency,  but  of  intersection,  and  these  sur- 
faces, simultaneously  swept  into  existence  by  the  auxiliary  hyperboloid, 
are  instantaneously  swept  out  again  by  each  other.  That  this  would 
occur  in  the  special  case  in  which  the  pitch  hyperboloids  retain  their 
limiting  forms,  the  one  being  a  cone  and  the  other  a  plane  disc  or 


A   KEW   THEORY — OLIVIER'S   INVOLUTES. 


253 


rather  annulus,  as  in  Fig.  101,  was  pointed  out  by  the  author  in  a 
paper  published  some  years  since  ;  *  but  not  at  the  time  of  that  writing, 
nor  until  the  present,  was  it  perceived  that  the  analogy  fails  in  gen- 
eral, and  that  this  whole  method  of  constructing  teeth  for  skew  wheels 
is  radically  wrong. 

382.  A  Practicable  Method. — A  different  line  of  procedure  must, 
therefore,  be  sought ;  and  a  basis  of  operations  which  will  lead  to 
reasonable  results  is  found  in  a  peculiar  property  of  the  involute,  first 
pointed  out  by  M.  Theodore  Olivier. 

In  Fig.  236,  AB  is  the  common  tangent  of  the  two  circles  whose 
centres  are  C'  and  />,  and  is,  therefore,  the  locus  of  contact  of  the 
involutes  O'P,  PHy  shown  in  contact  at  P.  Let  the  first  circle  be 
now  revolved  on  the  line  AB, 
as  on  a  hinge,  until  its  centre 
reaches  the  position  C9  the 
circle  then  appearing  as  an 
ellipse,  and  the  tooth  taking 
the  position  OP.  The  axes 
will  then  lie  in  different 
planes,  but  it  is  evident  that 
OP  will  still  drive  PH  exactly 
as  before,  AB  being  the  locus 
of  contact,  since  it  is  the  inter- 
section of  the  planes  of  the  base 
circles,  whatever  their  inclina- 
tion to  each  other. 

Therefore  the  action  of  the 
fronts  of  the  teeth  is  not  af- 
fected by  the  change  in  the 
position  of  the  originally  par- 
allel axes  ;  but  with  regard  to 
the  backs  the  case  is  quite  different.  Their  locus  of  contact,  when  the 
circles  were  in  the  same  plane,  was  the  other  common  tangent  EF9 
which  cuts  AB  at  G,  and  it  will  readily  be  seen  that  the  two  points  of 
the  backs  of  the  teeth  which  would  meet  at  G  in  the  first  place,  will 
do  so  still,  but  no  other  points  of  those  curves  will  come  into  contact 
at  all.  Consequently,  regarding  the  wheels  as  of  no  sensible  thick- 
ness, we  perceive  that  either  of  them  can  drive  continuously  in  only 
one  direction. 

383.  Of  this  combination  Prof.  Willis  remarks,  f 

*  Scientific  American  Supplement,  Nos.  174,  176  and  178. 
f  Principles  of  Mechanism,  p.  152. 


FIG.  236. 


254 


A   NEW  THEOKY — OLIVIEK'S   INVOLUTES. 


"Involute  wheels,  therefore,  maybe  employed  to  communicate  a 
constant  velocity  ratio  between  axes  that  are  inclined  at  any  angle  to 
each  other,  but  which  do  not  meet.  But  the  demonstration  supposes 
the  wheels  to  be  very  thin,  since  they  coincide  with  the  planes  that 
intersect,  and  the  invariable  points  of  contact  are  situated  in  this 
line  of  intersection.  The  edge  of  one  of  the  wheels  must  be  in  prac- 
tice rounded  so  that  it  may  touch  the  other  teeth  in  a  point  only." 

This,  however,  is  not  the  only  expedient  for  securing  substance  or 
body  for  the  teeth,  although  it  is  strictly  true  if  both  the  curves  are 
involutes.  But,  retaining  one  of  them,  it  is  possible  so  to  modify  the 
other,  when  sensible  thickness  is  given  to  the  wheels,  that  the  locus 
of  contact  shall  not  coincide  ^vith  the  intersection  of  the  two  planes, 
and  to  form  teeth  which  shall  work  together  in  contact  along  a  right 
line. 

384.  In  Fig.  237,  let  AB,  the  common  tangent  to  the  bases  of  the 


w 


FIG.  237. 


two  cones,  be  parallel  to  the  common  perpendicular  of  their  axes, 
whose  extremities,  V  and  W,  are  the  vertices.  In  the  tangent  plane 
thus  determined,  draw  EF  parallel  to  AB  ;  it  will  be  tangent  to  the 
two  circles  cut  from  the  cones  by  planes  parallel  to  their  bases,  and 
dividing  proportionally  both  the  axes  and  the  elements. 

The  involutes  PG,  PH,  of  the  bases,  will  work  together  as  in  Fig. 
236  ;  so  too  will  those  of  the  other  sections,  shown  in  contact  at  R. 
These  curves  may  be  considered  as  traced,  each  in  its  own  plane,  by 
marking  points  at  P  and  R,  drawn  along  by  two  cords,  AB  and  EF, 
while  the  cones  turn  as  indicated  by  the  arrows  ;  in  which  motion 
the  linear  velocities  of  the  marking  points  are  directly  proportional 


A   HEW  THEORY  —  OLIVIER'S  INVOLUTES.  255 

to  the  radii  of  the  bases  of  either  frustum,  and  these  to  their  distances 
from  the  vertex,  measured  either  on  the  axis  or  on  an  element. 

Draw  PR  and  produce  it  to  cut  FTFin  0;  then  if  P  move  to  a 
new  position  /or  L,  the  corresponding  position  of  R  will  be  M  or  N, 
such  that 


AV  -  BW  - 

EV    ~~  FW'~~  RO* 

that  is  to  say,  the  point  0  and  the  two  marking  points  will  always 
lie  in  one  straight  line. 

385.  The    Tooth-surface    a    Single-curved  One.  —  Consequently  any 
number  of  tracing  points  may  be  placed  between  P  and  0,  each  in 
like  manner  generating  a  pair  of  involutes  which  will  work  together 
as  in  the  preceding  figure  ;  the  point  0  itself  describes  two  circles, 
which  may  be  regarded  as  the  involutes  of  their  centres  Fand  W. 
And  the  straight  lines  joining  the  corresponding  points  of  the  invo- 
lutes belonging  to  either  cone  constitute  a  surface  ;  which  may  also 
be  generated  by  sliding  a  right  line  upon  any  three  of  these  involutes 
as  directrices. 

In  the  latter  mode  of  generation,  every  point  of  the  right  line  is  at 
any  instant  moving  in  the  same  direction,  because  the  tangents  to  all 
the  involutes  are  parallel  :  but  the  upper  end  is  moving  faster  than 
the  lower  end.  Therefore,  the  consecutive  elements  intersect  each 
other  two  and  two,  and  the  surface  is  not  a  warped  one,  but  single 
curved,  and  the  plane  tangent  to  it  at  any  point  is  tangent  to  it  all 
along  a  right  line. 

But  it  does  not  follow,  nor  is  it  true,  that  the  two  surfaces  thus 
simultaneously  generated  by  the  line  PO  will  be  tangent  to  each 
other.  The  plane  tangent  to  either  at  any  point  is  determined  by 
the  rectilinear  element,  and  the  tangent  to  the  involute,  which  pass 
through  that  point.  But  the  two  involutes  generated  as  above  lie  in 
different  planes,  their  tangents  have  different  directions,  and  in  con- 
sequence the  plane  tangent  to  one  surface  is  not  tangent  to  the  other. 
Retaining  one  of  them,  however,  it  is  possible  to  construct  another 
which  shall  be  tangent  to  it  in  each  of  the  different  positions,  by  a 
process  analogous  to  that  of  deriving  a  conjugate  tooth  from*  a  given 
form. 

386.  Adaptation  to  Hyperboloidal  Wheels.  —  The  application  of  this 
in  the  case  of  hyperboloidal  wheels  will  be  understood  by  reference  to 
Fig.  233  ;  if  through  C  and  D  two  lines  be  drawn  parallel  to  the 
common  element  AB,  these  lines,  by  revolving  about  the  vertical  and 
inclined  axes  respectively,  will  generate  two  cones  situated  relatively 


256 


ADAPTATION-   TO    SKEW   WHEELS. 


to  each  other  as  in  Fig.  237,  and  the  tooth-surfaces  for  the  wheels 
may  be  formed  as  above  explained.  Those  for  the  first  wheel,  being 
composed  of  true  involutes,  may  be  constructed  without  reference  to 
the  second,  as  shown  in  Fig.  238,  which  is  drawn  without  regard  to 
practical  conditions  or  proportions,  for  the  purpose  of  exhibiting  dis- 
tinctly some  peculiarities  which  must  be  taken  into  account  in  laying 
out  such  teeth  for  actual  use. 
Let  KIM  be  the  upper  base  and  fiOFihe  gorge  circle  of  the  hyper- 


Z  X     Y 


FJGS.  238  and  239. 


boloid,  whose  generatrix,  OM,  is  parallel  to  the  vertical  plane,  K8,  con- 
taining'the  axis  C.  Through  M  draw  a  perpendicular  to  OM,  pierc- 
ing this  plane  at  A' ;  then  the  circle  about  C  through  A1  will  be  the 
base  of  the  cone.  On  A  'M  produced  take  any  point  P,  and  let  PO 
be  the  position  of  the  generatrix  of  the  tooth-surface  at  the  beginning 
of  the  action.  The  wheel  turning  as  shown  by  the  arrow,  let  the 
action  continue  until  a  marking  point  at  P  reaches  A',  tracing  the 
involute,  PA,  whose  position  at  the  end  is  P'A.  Any  other  point  R 


TEETH    VANISH   AT   THE    GORGE.  257 

on  P  Owill  meantime  trace  another  involute,  RB,  whose  final  position 
is  R'B',  while  0  will  trace  the  arc  00' ;  and  the  final  position  of  the 
generatrix  is  OB1  A'.  When  P  has  advanced  to  M,  R  will  have 
reached  N,  and  OM  is,  it  will  be  noted,  tangent  to  all  the  involutes 
traced  by  points  upon  PO. 

387.  The  Teeth  Vanish    at    the   Gorge  Plane.— In  Fig.    239,   MO 
is  the  generatrix,  and  SVFX  the  meridian  outline,  of  the  pitch  hy- 
perboloid. 

Now,  considering  first  the  part  which  lies  above  the  gorge  plane 
EF,  we  perceive  in  Fig.  238  that  the  outer  points  of  the  teeth,  as  P, 
R,  lie  in  the  line  PO,  whose  least  distance  from  the  axis  is  CG,  and 
G  is  below  the  plane  EF.  Similarly,  the  lowest  points,  A',  B',  lie  in 
a  line,  A'O,  of  which  the  point,  If,  nearest  to  the  axis,  lies  above  that 
plane.  These  two  lines  generate  two  other  hyperboloids,  whose 
meridian  outlines  in  Fig.  239  are  respectively  TWFtind.  MHF. 

The  teeth  then  vanish  at  the  gorge  circle,  but  if  the  generatrix  be 
continued  below  it,  they  will  reappear  ;  and  their  highest  and  lowest 
points  will  also  lie  upon  an  exterior  and  an  interior  hyperboloid.  It 
is  here  to  be  noted,  that  it  is  not  necessary  thus  to  prolong  the  same 
generatrix ;  but  if  it  be  done,  as  in  these  diagrams,  the  interior  hy- 
perbola, FGZ,  will  be  a  continuation  of  the  exterior  one,  TWF,  and 
FY  an  extension  of  MHF.  In  this  case  it  is  also  necessary  to  make 
the  angle  POM  in  Fig.  238  equal  to  the  angle  MO  A,  if  it  be  required 
that  the  teeth  of  the  upper  and  lower  portions  shall  be  symmetrically 
disposed,  so  that  for  example  Q'J'  may  be  equal,  similar,  and  opposite 
to  P'A'.  Such  symmetry,  though  not  essential,  is  obviously  desira- 
ble if  the  wheels  are  to  be  used  in  double  pairs  (161)  ;  but  the 
angles  just  mentioned  need  riot  be  made  equal ;  and  if  not,  the  result 
Avill  be  that  in  Fig.  229,  although  TWF  and  .FG^will  be  respect- 
ively similar  to  FY  and  FHM,  they  will  be  parts  of  different  hyper- 
bolas. 

388.  Limit  to  the  Prolongation  of  the  Tooth. — It  is  to  be  understood 
that  in  Fig.  238,  as  in  previous  cases,  the  vertical  plane  containing 
the  axis  C  is  parallel  to  the  axis  of  the  engaging  wheel,  the  point  0 
lying  upon  the  common  perpendicular.     If  then  the  angle  MO  A'  be 
assumed,  the  point  A'  in  which  the  generatrix  of  the  tooth-surface 
pierces  that  plane  limits  the  distance  of  the  upper  base  from  the  gorge 
circle.     For  the  transverse  plane  through  that  point  cuts  from  the 
fundamental  cone  a  circle  to  which  PMA'  is  tangent,  so  that  A'  is  the 
root  of  the  involute  traced  by  the  marking  point  in  that  plane  ;  and  a 
marking  point  in  any  higher  plane,  in  the  prolongation  of  OP,  would 
advance  beyond  the  point  of  tangency  between  its  path  and  the  corre- 

17 


258 


CONSTRUCTION   OF   BACK    OF  TOOTH. 


spending  section  of  the  cone,  and  thus  begin  to  trace  the  reverse 
branch  of  the  involute  of  that  circle,  which  is  clearly  inadmissible. 
Conversely,  if  the  distance  from  the  gorge  be  assumed,  the  point  A' 
is  thereby  determined,  which  in  turn  fixes  the  greatest  possible  value 
of  the  angle  A'OM. 

"We  have  thus  far  referred  only  to  the  lowest  acting  points  of  the 
teeth  which  lie  upon  the  interior  hyperboloid  in  Fig.  239.  But  it 
may  be  necessary  to  cut  more  deeply  into  the  blank  in  order  to  pro- 
vide clearing  spaces  for  the  passage  of  the  teeth  of  the  engaging  wheel. 
The  involutes  are  then  to  be  continued  as  far  as  required  below  the 
last  acting  points  ;  should  the  root  of  the  involute  have  been  already 
reached  in  any  transverse  plane,  as  at  A'  in  Fig.  238,  the  continuation 
in  that  plane  will  be  a  radial  line. 

389.  Construction  of  the  Back  of  the  Tooth. — The  complete  working 
teeth  must  be  provided  with  backs  as  well  as  fronts  ;  and  we  have  now 
to  consider  the  results  of  this  operation.  In  Fig.  240  we  have  a 


FIG.  240. 

single  tooth  upon  a  larger  scale,  the  curves  A'P',  B'R,  correspond- 
ing to  those  similarly  lettered  in  Fig.  238.  For  our  present  purpose 
we  may  disregard  the  clearing  space,  and  suppose  the  depths  deter- 
mined by  the  points  A',  B',  to  be  sufficient.  Considering  A'P',  then,  as 
the  front  of  the  tooth,  its  back  is  to  be  a  similar  involute  in  the  same 
plane,  and  for  illustration  we  will  assume  the  pitch  to  be  such  as  to 
make  the  tooth  pointed,  as  shown  ;  then  the  radius  through  P'  will 
bisect  the  arc  A'D,  which  measures  the  thickness  of  the  tooth  on  the 
circle  through  A. 

Make  the  angle  OCG  equal  to  the  angle  A  CD  ;  then  the  arc  QG  is 
the  thickness  of  the  tooth  at  the  gorge  circle,  and  the  lowest  (acting) 


CONSTRUCTION   OF  BACK   OF  TOOTH.  259 

points  of  the  backs  will  lie  upon  the  right  line  GD.  Thus,  a  circle 
through  B',  about  C ',  cuts  GD  in  E,  and  B'E  is  the  thickness  of  the 
tooth  in  the  plane  containing  the  involute  B'R.  The  radius  which 
bisects  B'E,  cuts  B'R  at  K,  through  which  point  passes  the  reverse 
involute,  EK,  or  back  of  the  tooth  in  that  plane  ;  in  alike  manner  we 
determine  the  section  HLF  of  the  complete  tooth  by  a  plane  still 
nearer  the  gorge,  and  so  on  indefinitely. 

390.  The  Extension  of  the  Tooth  Practically  Limited  in  Both  Direc- 
tions.— It  will  be  observed  that  the  points  of  these  sections  become 
more  and  more  obtuse,  and  the  depths  less  and  less,  as  we  approach 
the  gorge  plane.     And  in  that  plane,  bisecting  OGby  the  radius  CJ, 
the  front  of  the  tooth  is  the  arc  OJ,  and  its  back  is  the  arc  GJ ";  these 
being,  as  previously  suggested,  the  involutes  of  the  centre,  C. 

If  then  we  make  the  section  of  the  tooth  by  the  plane  farthest  from 
the  gorge  pointed,  we  perceive  that  the  fronts  of  the  sections  nearer 
to  the  gorge  cannot  be  extended  to  the  full  heights,  as  /,  R,  found 
as  in  Fig.  238,  since  they  are  intersected  by  the  backs  before  those 
heights  are  reached.  And  a  sufficient  number  of  these  intersections, 
as  L,  Ky  being  determined,  a  line,  JLKP',  is  drawn  through  them, 
which,  by  revolving  round  the  axis,  (7,  will  generate  a  surface  which  is 
evidently  the  correct  blank  for  the  wheel  upon  the  above  supposition. 

The  meridian  outline  of  this  surface  is  the  curve  TVF,  in  Fig.  239, 
which,  it  is  to  be  noted,  intersects  the  pitch  hyperboloid  at  V.  But 
OM,  of  Figs.  238  and  239,  the  generatrix  of  that  hyperboloid,  is  also 
a  line  of  the  front  tooth-surface.  This  is  now  cut  off  at  V  by  the 
transverse  plane  through  V,  and  it  is  clear  that  between  this  plane 
and  the  gorge  the  teeth  cannot  possibly  engage  with  each  other. 
Practically,  then,  they  should  terminate  still  farther  from  the  gorge 
circle. 

391.  Suppose  now,  however,  that  the  pitch  is  so  much  increased 
that  the  section  of  which  B'R'  is  the  front  may  be  carried  up  to  the 
full  height  and  made  pointed  at  R.     The  reverse  involute  through  R 
will  cut  the  circle  through  B'  at  a  point  JV;  the  angle   OCG  must 
then  be  made  equal  to  that  measured  by  the  arc  B'N9  and  a  straight 
line  from  this  new  position  of  G,  through  N,  of  which  NQ  is  a  por- 
tion, will  contain  the  lowest  acting  points  of  the  backs  under  the  new 
condition,  as   did  GD  under  the  old  one.     Below  R  the  state  of 
things  will  be  analogous  to  that  which  previously  existed  below  P'  • 
and  we  may  proceed  to  construct  as  before  a  curve  corresponding  to 
JLCP',  and  to  ascertain  the  limit  beyond  which  the  teeth  cannot  be 
continued  in  the  direction  of  the  gorge. 

But  above  R  we  shall  have,  in  the  first  place,  a  right  line,  R'P1, 


2GO  ACTIOK   OF   BACK   OF  TOOTH. 

which  is,  of  course,  a  portion  of  P'O'  of  Fig.  238,  the  generatrix  of 
the  exterior  hyperboloid,  TWF,  of  Fig.  239,  which  is  now  the  form  of 
the  blank.  In  the  next  place  the  tooth  above  R  will  not  be  pointed, 
but  will  have  a  sensible  thickness  at  the  top.  Thus,  the  arc  A'Q 
being  made  equal  in  circular  measure  to  B'N  and  QT  similar  and 
opposite  to  A  P',  the  tooth  outline  on  the  upper  plane  is  bounded  at 
the  top  by  the  arc  P'T.  Other  planes  perpendicular  to  the  axis  may 
be  passed  at  any  convenient  intervals,  and  the  arc  limiting  the  section 
of  the  tooth  at  the  top  being  found  in  like  manner  for  each,  the  line 
R'Tis  determined,  which  is  the  outer  edge  of  the  back  of  the  tooth, 
and  lies,  of  course,  upon  the  hyperboloid  generated  by  R'P' .  It  is 
not  a  straight  line,  but  in  most,  if  not  all,  practical  cases,  its  curvature 
within  the  limits  employed  will  be  very  slight. 

392.  Nature  and  Action  of  the  Back  of  the  Tooth. — Evidently,  when 
we  thus  assume  any  plane  as  the  lower  base  of  the  frustum  to  be  used 
for  a  wheel,  we  need  not  make  the  section  of  the  tooth  by  that  plane 
pointed  ;  for  instance,  we  might  have  made  B'M  greater,  so  that  the 
lower,  as  well  as  the  upper,  section  would  have  been  blunted.  This, 
however,  would  not  in  the  least  affect  the  mode  of  proceeding,  nor  the 
nature  of  the  surface  which  forms  the  back  of  the  tooth.  This  sur- 
face can  not  be  generated  as  was  that  forming  the  front,  as  will  be 
seen  from  the  consideration  that  the  tangents  to  the  involutes  at  D, 
E,  and  F,  in  Fig.  240,  are  not  parallel.  Therefore  it  is  not  of  single 
curvature ;  it  is  not  of  double  curvature,  for  it  contains  the  right  line 
GD ;  it  is,  then,  a  warped  surface,  and  generated  by  sliding  the  right 
line  upon  any  three  of  the  involutes  as  directrices. 

Nor  does  it  work  in  right-line  contact  with  the  engaging  tooth  of 
the  other  wheel.  Supposing  the  latter  for  the  moment  to  be  of  the 
same  nature,  it  will  be  seen,  by  reference  to  Fig.  236,  that  the  only 
point  of  contact  between  the  involutes  of  any  two  transverse  sections 
of  the  fundamental  cones  which  have  a  common  tangent  (which  may 
be  called  conjugate  sections),  is  found  by  revolving  either  of  those 
circles  about  that  line  until  it  lies  in  the  plane  of  the  other,  and  then 
drawing  the  line  of  centres,  which  will  cut  the  tangent  at  the  point 
in  question. 

Were  the  wheels  made  up  of  laminae,  each  acting  upon  its  conjugate 
only  in  this  manner,  we  should  have  a  series  of  such  contacts,  the  first 
occurring  in  the  planes  nearest  the  gorge,  the  last  in  those  most 
remote  ;  the  teeth  touching  each  other  in  a  single  point,  which  would 
travel  endlong,  in  a  line  parallel  and  near  to  the  common  element  of 
the  pitch  hyperboloids  ;  very  much  as  in  the  case  of  twisted  spur 
or  bevel  wheels. 


BACK   OF   CONJUGATE   TOOTH.  261 

393.  Peculiar  Action  of  the  Backs  of  the  Teeth. — This,  however,  is 
not  the  case,  as  the  supposed  laminae  are  not  capable  of  such  independ- 
ent action.     Let  a  conjugate  pair  at  any  instant  touch  each  other  as 
above  ;  then  at  the  next  instant,  the  consecutive  point  of  one  section 
will  come  into  contact  with  the  engaging  tooth  at  a  point  not  situated 
in  the  conjugate  plane  but  in  a  plane  beyond.     But  since  the  above 
described   contacts    between    conjugate    planes  occur  at    successive 
instants,  we  have  in  effect  that  successive  phases*  of  the  action  are 
simultaneously  represented,  the  result  being  that  the  two  teeth  at  any 
given  instant  touch  each  other  along  a  curved  line  of  limited  length  ; 
each  point  of  which  during  the  action  travels  along  the  tooth  from 
end  to  end  in  the  manner  previously  explained. 

394.  Construction  of  the  Back  of  the  Conjugate  Tooth.— Under  these 
circumstances,  if  the  back  of  the  tooth  of  one  wheel  be  made  up  of 
the  reverse  involutes,  as  at  first  supposed,  it  is  not  demonstrable  that 
the  back  of  the  engaging  tooth  should  be,  but  its  proper  form  may  be 
determined  n,s  follows :  pass  any  plane  perpendicular  to  the  axis  of 
the  second  wheel,  and  upon  it  draw  the  outline  of  the  section  cut  by 
it  from  the  assumed  tooth  of  the  first.     Rotate  this  plane  through  a 
small   angle,  and  turn   the   first  wheel   through  the  corresponding 
angle,  determined  by  the  given  velocity  ratio.     The  outline  of  the  new 
section  of  the  given  tooth  is  now  to  be  traced  upon  the  plane,  and  the 
process  repeated  until  tho  section  of  the  conjugate  tooth,  which,  of 
course,  is  the  envelope  of  the  various  curves  thus  traced  upon  the  perpen- 
dicular plane,  is  mapped  out  with  the  desired  degree  of  accuracy.     A 
new  transverse  plane  is  now  to  be  passed,  and  the  same  series  of  opera- 
tions again  performed  ;  and  this  once  more  repeated,  gives  us  finally 
three  directrices,  upon  which  a  right  line  being  made  to  slide  will 
generate  the  back  of  the  required  conjugate  tooth. 

395.  Construction  of  the  Front  of  the  Conjugate  Tooth. — The  front 
also  of  this  conjugate  tooth,  as  stated  in  (385),  will  be  different  from 
that  of  the  first  wheel,  whose  generation  was  illustrated  in  Fig.  238. 
Referring  to  that  diagram,  it  is  seen  that  in  every  position  of  the  gen- 
eratrix, the  tangents  to  all  the  involutes  are  parallel  to  OM ;  and  they 
are  all  parallel  to  the  gorge  plane.     Therefore  OM  is  the  trace  upon 
the  gorge  plane   of  the  plane  tangent  to  the  tooth-surface  at  any 
phase  of  the  action  ;  the  latter  plane  turning  upon   OM  as  upon  a 
hinge,  during  the  rotation  of  the  wheel.     And  in  each  of  its  positions 
it  will  cut  a  right  line  from  any  transverse  plane  of  the  second  wheel, 
which,  meantime  rotating  about  its  own  axis,  will  thus  have  traced 
upon  it  a  series  of  lines,  to  which  the  front  outline  of  the  section  of 
the  conjugate  tooth  by  that  plane  mnst  be  tangent. 


262 


FROKT   OF   CONJUGATE   TOOTH. 


Thus,  in  Fig.  241  O'C'  is  the  vertical  axis  of  the  first  wheel,  corre- 
sponding to  C  of  Fig.  238,  O'M'  corresponds  to  OM  of  the  same 
diagram,  O'D'  is  the  axis  of  the  second  wheel,  and  the  right-hand 
portion  of  this  figure  is  a  projection  upon  a  plane  perpendicular  to 
the  inclined  axis  ;  in  which  PABisihe  path  of  a  marking-point  which 
generates  an  involute  pertaining  to  the  first  wheel,  this  path  corre- 
sponding to  PMA'  of  Fig.  238.  The  arrow  indicating  the  direction  of 
the  rotation,  P  is  already  the  lowest  possible  point  of  the  acting  face 
of  the  tooth  of  the  second  wheel,  and  PM  is  the  position  of  the  plane 
tangent  to  the  tooth  of  the  other  at  this  phase.  When  P  has  moved 
to  A  this  plane  will  have  the  position  A M,  and  the  radius  DP  will  be 
found  at  DE,  the  arc  PE  being  equal  to  PA.  So  when  P  reaches  B, 
the  plane  will  appear  as  BM,  and  DF  will  be  the  new  position  of  DP, 
the  arc  PF  being  equal  to  PB,  Thus  the  outline  of  the  tooth  being 


M 


FIG.  241. 


tangent  to  the  trace  of  the  plane  in  each  of  its  successive  positions, 
we  are  enabled  to  map  out  its  form  as  shown  at  FB. 

Evidently  this  curve  is  not  a  true  involute,  which  could  be  produced 
only  on  the  supposition  that  the  tangent  plane  should  always  be  par- 
allel to  AM ;  but  in  most  practical  cases  the  angle  PMB  will  be  small, 
and  the  deviation  of  the  tooth-outlines  from  the  involute  form  will  be 
comparatively  slight. 

396.  Different  Actions  on  Opposite  Sides  of  the  Gorge. — A  singular  fact 
in  relation  to  the  teeth  thus  constructed  is  that  on  opposite  sides  of  the 
gorge  plane  the  fronts  and  backs  are  transposed,  as  will  be  seen  by  refer- 
ence to  Fig.  238  ;  so  that  if  a  double  pair  of  such  wheels  be  employed, 
turning  in  a  given  direction,  the  work  of  one  pair  will  be  done  by  the 
single  curved  surfaces  of  the  teeth,  that  of  the  other  pair  by  the 
warped  surfaces,  the  distribution  being  reversed  if  the  directions  of 
the  rotations  be  changed. 


APPEARANCE  OF  COMPLETE  WHEEL. 


263 


Due  attention  should  be  paid  to  this  point  if  a  single  pair  only  is  to 
be  used,  and  unless  the  arrangement  of  other  parts  of  the  machine 
prevents,  the  wheels  should  be  placed  on  that  side  of  the  gorge  plane 
which  will  secure  the  advantage  of  having  the  acting  surfaces  of  the 
teeth  touch  each  other  throughout  their  length. 

397.  Practically,  then,  this  construction  enables  us  to  make  a  pair 
of  skew  wheels  which  in  one  direction  of  the  driver's  rotation  will 
work  in  right-line  contact.  The  location  of  the  upper  bases  of  the 
hyperboloidal  frusta  being  arbitrary,  they  should  be  placed  as  far  as 


TJ 

:^> 

FIG.  242. 

possible  from  the  gorge  planes,  in  order  to  reduce  the  transverse 
obliquity  to  the  lowest  limit. 

The  blanks  should,  of  course,  be  terminated  by  cones  normal  to  the 
pitch  surfaces,  as  previously  explained  ;  the  traces  of  the  teeth  upon 
these  cones  are  most  readily  found  by  passing  a  series  of  transverse 
planes,  and  drawing  in  each  the  outlines  of  the  tooth  ;  which  will 
intersect  the  circles  cut  from  the  cones  in  points  of  the  required 
curves. 


264  THE  ANALOGY  CORRECTED  AND  COMPLETED. 

The  appearance  of  the  complete  wheel  as  thus  laid  out  is  shown  in 
Fig.  242,  which  represents  the  larger  of  a  pair  having  respectively  48 
and  32  teeth,  the  angle  between  the  projections  of  the  axes  being  60.° 
On  the  right  is  shown  the  development  of  the  tooth-outline  on  the 
outer  normal  cone  of  the  larger  wheel,  the  corresponding  development 
for  the  smaller  one  being  given  on  the  left.  These  outlines,  it  will 
be  observed,  are  by  no  means  symmetrical,  owing  to  the  transverse 
obliquity  of  the  teeth,  although  in  one  wheel  the  sections  by  planes 
perpendicular  to  the  axis  may  be  symmetrical  with  respect  to  the  cen- 
tral radii. 

398.  It  will  be  apparent  on  reflection  that  the  analogy  between 
conical  and  hyperboloidal  wheels,  in  order  to  be  perfect,  requires  that 
the  teeth  of  the  latter  should  vanish  at  the  gorge,  as  do  those  of  the 
former  at  the  vertex.    It  is  true  that  models  have  been  made,  as,  for 
instance,  those  by  Schroeder  of  Darmstadt,  in  which  teeth  of  sensible 
magnitude  arc  given  to  wheels  whose  mid-planes  coincide  with  the 
gorge   circles  of  hyperboloids,  whose  elements  are  represented  by 
wires  passing  through  the   teeth  ;   it  is  also  true  that  these  wheels 
work,  transmitting  the   rotation  with   perfect  constancy  of  velocity 
ratio. 

But  it  does  not  follow  that  these  are  true  skew  teeth,  nor  that  they 
work  in  right-line  contact  at  all  on  either  side.  These  wheels,  as 
made,  are  thin  ;  and,  as  will  be  seen  presently,  screw  teeth  formed 
upon  pitch  cylinders  tangent  to  these  hyperboloids  at  the  gorge 
circles  will  curve  so  little  in  the  small  portion  used,  that  it  would  be 
difficult  by  mere  inspection  to  detect  the  curvature  or  to  ascertain 
whether  contact  existed  in  more  than  one  point.  And  in  fact  such 
teeth  are  very  often,  if  not  always,  actually  made  by  means  of  a  mill- 
ing cutter  travelling  in  the  direction  of  the  tangent  to  the  helix  at  the 
mid-plane  of  the  wheel,  without  rotating  the  blank  during  the 
operation. 

But  these  wheels  cannot  be  made  of  any  considerable  length  in  the 
direction  of  the  axis,  and  the  suspicion  of  their  identity  with  those  in 
the  models  above  mentioned  has  yet  to  be  removed  by  the  production 
of  a  pair  in  which  straight-line  teeth  extend  past  the  gorge  circle's 
from  end  to  end  of  long  hyperboloids. 

399.  Skew-bevel  wheels  are  not  often  met  with  in  practice.     The 
usual  expedient,  when  two  axes  lie  in  different  planes,  is  to  introduce 
a  counter-shaft,  whose  axis  intersects  both  the  others,  and  to  use  two 
pairs  of  bevel  wheels.     And  when  they  are  at  a  great  distance  from 
each  other  this  may  be  unavoidable  ;  but  if  they  be  not,  there  can  be 
no  question  that  the  loss  of  power  due  to  the  imperfect  rolling  of  the 


TWISTED   SKEW   GEABING. 


265 


\ 


pitch  surfaces,  and  the  transverse  obliquity  of  the  teeth, with  a  pair 
of  properly  constructed  skew  wheels  would  in  many  cases  be  less  than 
that  incurred  when  the  arrangement  above  mentioned  is  adopted  ;  to 
say  nothing  of  the  superiority  of  the  single  pair  in  respect  to  neatness, 
lightness,  and  compactness. 

400.  Twisted  Skew  Wheels. — If  we  suppose  a  pair  of  tangent  hyper- 
boloids  to  be  made  up  of  a  series  of  transverse  laminae,  it  is  clear  that 
by  twisting  them  uniformly,  as  in 
Fig.  243,  the  rectilinear  elements 
will  be  changed  into  hyperboloidal 
helices,  while  the  surfaces  still  touch 
each  other  along  a  right  line  as 
before.  And  had  teeth  been  added 
previously  to  the  twisting,  then,  as 
was  the  case  with  the  spur  and  the 
bevel  wheels,  these  teeth  would  con- 
tinue to  act  with  the  proper  velocity 
ratio.  . 

This  at  first  glance  appears  a  useless 
addition  to  an  already  ample  degree 
of  complexity,  but  upon  closer  exam- 
ination it  will  be  seen  that  such  twisted 
teeth  can  actually  be  made  more  easily 
than  straight  ones.  The  hyperbo- 
loidal helix  can  be  traced  by  suppos- 
ing a  marking  point  to  travel  uni- 
formly along  an  element,  while  the 
surface  turns  uniformly  upon  its  axis. 
And  if  the  teeth  begin  and  end  their 
contact  upon  that  line,  as  in  Dr. 
Hooke's  spiral  wheels,  this  will  be  the 
only  line  of  their  surfaces  whose  form  is  of  essential  importance. 

Practically,  therefore,  it  is  requisite  merely  to  arrange  proper  mech- 
anism for  simultaneously  moving  a  milling  cutter  along  the  line  of  an 
element  of  the  pitch  surface,  and  rotating  the  blank  upon  its  axis, 
both  motions  to  be  uniform.* 

The  same  mode  of  proceeding  holds  good  when,  as  in  Pig.  101,  the 
hyperboloids  retain  the  limiting  forms  of  a  cone  and  a  plane,  for  if 
we  imagine  the  disc  to  be  made  up  of  as  many  concentric  rings  as 
there  are  laminse  in  the  cone,  each  one  will  be  driven  round  by  the 
twisting  of  the  latter,  so  that  the  original  line  of  tangency  on  the 

*  Scientific  American  Supplement,  No.  178. 


\ 


FIG.  243. 


266 


TEETH   OF   SCREW   WHEELS. 


plane  will  be  distorted  into  a  curve  of  a  spiral  form,  which  may  also 
be  traced  as  above  described  by  the  uniform  motion  of  a  marking 
point  along  the  right  line  of  contact,  while  the  disc  rotates  uniformly 
about  its  axis. 

The  teeth  thus  formed,  it  is  evident,  will  have  but  a  single  point  of 
contact,  which  will  travel  along  the  common  element  of  the  pitch 
surfaces,  just  as  in  the  case  of  other  twisted  wheels  ;  but  this  is  suffi- 
cient to  make  the  action  continuous,  if  the  amount  of  twist  in  the 
length  of  the  tooth  be  a  little  greater  than  the  pitch  ;  and  the  action 
will  be  peculiarly  smooth,  since  the  amount  of  sliding  friction  will  be 
no  greater  than  that  between  the  pitch  surfaces. 

The  Teeth  of  Screw   Wheels. 

401.  The  most  common  example  of  Screw  Gearing  is  the  arrangement 
familiarly  known  as  the  Endless  Screw,  or  Worm  and  Wheel.  In  this 
case  the  axes  are  situated  in  planes  which  are  perpendicular  to  each 
other,  and  the  nature  of  the  action  will  be  readily  seen  by  inspection 
of  Fig.  244.  0  being  the  centre  of  a  pitch  circle,  and  TT  the  pitch 

line  of  a  rack,  let 
teeth  be  constructed 
of  any  of  the  forms 
proper  for  spur 
gearing.  In  the 
plane  of  the  paper 
draw  any  line  DD 
parallel  to  TT,  and 
taking  it  as  an  axis, 
let  the  outline  of 
the  rack  be  made 
the  meridian  sec- 
tion  of  a  screw 
whose  pitch  is  equal  to  that  of  the  rack  teeth.  This  screw  is  still  a 
rack,  and  if  moved  endlong  will  turn  the  wheel  ;  but  if  instead  of  this 
the  screw  itself  be  turned,  the  effect  will  be  precisely  the  same.  For, 
supposing  the  wheel  to  be  very  thin,  its  tooth  is  confined  between  the 
threads  of  the  screw,  all  of  whose  meridian  sections  are  alike,  but  each 
successive  one  is  in  advance  of  the  preceding,  so  that  when  the  screw 
has  made  one  revolution,  the  wheel-tooth  must  have  been  driven 
through  an  angle  measured  by  the  pitch  arc.  In  short,  the  screw  is  a 
rack  which  advances  by  rotation  ;  and  this  is  the  fundamental  princi- 
ple of  all  screw  gearing,  with  the  exception  of  one  combination,  which 
will  be  described  hereafter. 


244. 


NATURE  OF  THE  ACTION. 


267 


402.  Distinctive  Peculiarities  of  the  Action. — The  line  TT,  by  revolv- 
ing about  the  axis  DD,  generates  the  pitch  cylinder  of  the  screw, 
which  is  tangent  to  that  of  the  wheel  at  the  point  A. 

Three  characteristic  features  distinguish  the  action  from  that  of 
twisted  gearing,  viz.  : 

1.  The  velocity  ratio  is  independent  of  the  relative  diameters  of  the 
pitch  cylinders,  and  depends  wholly  upon  the  screw  pitch. 

2.  The  directional  relation  depends  upon  the  direction  of  the  twist ; 
the  screw,  turning  in  a  given  direction,  will  drive  the  wheel  one  way 
if  right-handed,  the  other  way  if  left-handed. 

3.  The  rotation  of  the  wheel  is  caused  solely  by  the  end  thrust  of 
the  screw. 

403.  Wheels  with  Similar  Transverse  Sections. — In  giving  sensible 
thickness  to  the  wheel,  we  may  proceed  as  follows  :  the  elements  of 
the  two  pitch  cylinders  which  pass  through  A  determine  the  common 
tangent  plane  represented  by  MN,  in  Fig.   245.     The  screw  helix 
through  A  will  develope  upon  this  plane  into  a  right  line,  which,  when 
the  plane  is  wrapped  upon  the 

pitch  cylinder  of  the  wheel,  will 
become  another  helix  lying  on 
that  surface  ;  these  two  helices 
will  be  either  both  right-handed 
or  both  left-handed.  Through 
each  point  in  the  outline  of  the 
wheel-tooth,  already  laid  out, 
draw  a  helix  of  the  same  pitch ; 
we  shall  thus  have  a  wheel  pre- 
cisely like  one  of  the  twisted  pair 
shown  in  Fig.  109,  all  the  trans- 
verse sections  being  similar.  The  thickness  of  this  wheel  is  to  be  de- 
termined only  by  considerations  relating  to  the  pressure  to  be  trans- 
mitted and  the  strength  of  the  material;  it  has  no  bearing  upon  the 
kinematic  action,  since  at  any  instant  each  tooth  touches  the  engaging 
screw-thread  only  in  a  single  point  in  the  original  transverse  plane 
through  the  axis  of  the  screw.  Therefore,  this  form  of  wheel  is  open 
to  the  objection  that  the  wear  will  be  comparatively  rapid,  being  con- 
fined to  one  line  upon  each  thread  and  each  tooth. 

404.  Close-fitting  Tangent  Screws. — A  worm  wheel  to  which  this 
objection  does  not  apply  can  be  practically  made  in  this  manner  :  an 
exact  copy  of  the  screw  in  steel  is  notched  and  hardened  so  as  to  be- 
come a  cutting  tool,  which  is  used  to  finish  the  teeth,  usually  roughly 
cut  upon  the  blank  with  an  ordinary  milling  cutter.     The  cutting 


M 


268  CLOSE-FITTING  TANGENT  SCREW. 

screw  is  often  made  to  drive  the  worm  wheel  during  this  operation  ; 
but  a  better  plan  is  to  have  the  wheel -blank  driven  at  the  proper 
speed  by  independent  means. 

When  this  method  is  adopted  it  is  necessary,  after  taking  one  cut, 
to  press  the  axes  nearer  together,  and  then  take  a  lighter  finishing  cut. 
Therefore,  an  involute  wheel-tooth,  working  with  a  straight-sided  slo- 
ping rack  tooth,  is  to  be  preferred,  because  this  change  in  the  position 
of  the  axis  does  not  affect  the  velocity  ratio. 

In  this  manner  a  perfect  worm  wheel  is  practically  made  with  great 
facility.  The  accurate  delineation  of  it  is  more  difficult  and  tedious, 
but  it  can  be  made  by  a  process  illustrated  on  the  right  in  Fig.  244, 
where  is  shown  a  section  by  a  plane  through  the  axis  of  the  wheel  per- 
pendicular to  that  of  the  worm.  LL  is  the  mid-plane  of  the  wheel, 
in  which  the  teeth  were  first  laid  out,  as  above  explained. 

405.  If  now  we  pass  any  plane  as  JV  parallel  to  LL,  it  will  cut  from 
the  screw  a  curved  line  ;  this  being  taken  as  a  rack  tooth,  the  form  of 
the  wheel  tooth  which  will  work  with  it  may  be  determined  by  the 
process  of  Fig.  162,  and  this  will  be  the  outline  of  the  wheel  tooth  in 
that  plane.     Another  parallel  plane  at  M  will  give  a  different  section 
of  the  screw,  from  which  we  derive,  as  before,  the  conjugate  form  of 
the  wheel  tooth,  and  this  may  be  repeated  as  many  times  as  is  deemed 
necessary. 

The  blank  for  the  wheel  is  usually  of  the  form  shown  ;  the  line  OR 
describes  a  cone,  from  which  the  parallel  planes,  M  and  N,  cut  circles, 
and  the  intersections  of  these  circles  with  the  outlines  of  the  wheel- 
tooth  in  these  planes  will  be  points  in  the  visible  contour  of  the  tooth. 

406.  Superior  Action  of  the  Close-fitting  Screw. — Not  only  are  all 
meridian  sections  of  the  screw  alike,  but  all  sections  by  planes  parallel 
to  and  equidistant  from  its  axis  are  alike.     The  whole  screw  being  a 
rack  which  advances  by  rotation,  it  is  clear  that  at  each  instant  there 
will  be  points  of  contact  not  only  in  the  plane  LL,  but  in  other  con- 
secutive planes  as  N,  M,  etc.     These  points  constitute  a  line  of  contact 
which,  though  not  a  true  helix,  will  evidently  be  a  line  of  double  cur- 
vature of  kindred  nature,  and  during  the  rotation  it  will  travel  along 
the  wheel-tooth  from  the  point  toward  the  root.     Between  each  tooth 
and  its  thread,  then,  we  have  contact  along  a  line,  and  the  wear  is 
distribute^,  over  a  surface. 

407.  Practical  Proportions. — Abstractly  considered,  both  the  diameter 
of  the  worm  and  the  number  of  teeth  in  the  wheel  are  optional.     But 
it  is  found  that  in  practice  the  results  are  not  satisfactory  if  the  wheel 
has  less  than  from  twenty-five  to  thirty  teeth  ;  with  a  straight-sided 
rack  and  involute  wheel-tooth,  the  obliquity  being  15°,  thirty-six  teeth 


PEACTICAL   PKOPOKTIOKS.  269 

will  give  a  total  angle  of  action  greater  than  twice  the  pitch,  with  the 
arc  of  recess  one  and  a  half  times  as  great  as  that  of  approach,  which 
is  all  that  could  be  desired.  This  is,  of  course,  for  a  single-threaded 
worm  ;  in  regard  to  its  diameter,  a  simple  and  a  good  practical  rule 
is  to  make  the  radius  of  the  blank  about  twice  the  pitch.  If  the  wheel 
be  a  simple  twisted  one  with  all  its  transverse  sections  alike,  its  blank 
will  be  a  cylinder,  whose  thickness  should  be  from  two  and  a  half  to 
three  times  the  pitch ;  if  it  be  cut  by  the  screw,  and  of  the  form 
shown  in  Fig.  244,  the  angle  ROP  may  be  made  from  60°  to  90°,  the 
thickness  being  as  just  given. 

408.  Physical  Embodiment  of  Sang's  Theory. — The  employment  of 
the  screw  to  cut  its  own  wheel  at  once  suggests  the  formation,  in  the 
same  manner,  of  guide  templates  suitable  for  use  in  connection  with 
the  pantagraphic  cutter  engine  of  Pratt  &  Whitney.     In  cutting  such 
a  template  the  screw  practically  and  automatically  executes  the  process 
of  finding  the  form  of  a  tooth  conjugate  to  that  of  a  given  rack,  as 
illustrated  in  Fig.  162. 

Now  if,  as  in  Fig.  244,  we  use  a  straight-sided  sloping  rack  tooth, 
the  result  will  be  the  formation  of  a  series  of  involute  templates  ;  if 
cycloidal  arcs  be  substituted  for  straight  lines,  as  in  Fig.  120,  we  shall 
have  a  set  of  templates  for  epicycloidal  teeth  with  a  constant  describ- 
ing circle.  But  the  outline  of  the  screw-thread  may  be  made  of  any 
other  reasonable  form  ;  and  according  to  Sang's  Theory  (283),  if  it  be 
bounded  by  any  four  similar  and  equal  curves  in  alternate  reversion, 
the  series  of  templates  produced  will  be  interchangeable,  and  thus 
by  means  of  the  pantagraphic  engine,  the  cutters  for  wheels  upon  any 
desired  basis  or  system  may  be  readily  and  accurately  duplicated. 

409.  Multiple-threaded  Screw  Wheels. — Thus  far  we  have  supposed 
the  screw  to  be  single-threaded,  with  a  pitch  equal  to  that  of  the  fun- 
damental rack  tooth. 

Now  the  helical  pitch  may  be  doubled, 
as  shown  in  Fig.  246  ;  this  will  double 
the  angular  velocity  of  the  wheel,  but 
if  no  other  change  be  made,  the  alternate 
teeth  only  will  come  into  action.  This 
difficulty  is  obviated  by  making  the  screw 
double-threaded,  as  in  Fig.  247  ;  which 
at  the  same  time  reduces  by  one  half  the 
pressure  upon  each  tooth. 

In  like  manner  we  may  make  the  heli- 
cal pitch  three,  four,  or  any  whole  num- 
ber of  times  as  great  as  the  tooth-pitch,  increasing  the  number  of 


270 


MULTIPLE-THKEADED   SCEEW   WHEELS. 


threads  accordingly,  and  taking  care  to  make  the  diameter  great 

enough  to  avoid  excessive  obliquity  of  action. 
And  in  this  way  we  may  give  to  the  screw  as  many  threads  as  there 

are  teeth  upon  the  wheel,  or  even  more  ;  the  combination  then  having 

but  slight  resemblance  to  the  single-threaded  endless  screw,  as  will  be 

seen  by  referring  to  Fig.  110,  which  repre- 
sents a  pair  of  screw  wheels  properly  so 
called. 

When  the  numbers  of  the  threads  and 
teeth  are  equal,  Prof.  Willis  states  that  the 
two  wheels  may  be  made  exactly  alike  ;  * 
this  we  imagine  to  be  a  mere  slip  of  the 
pen,  since  that  eminent  writer  was  the 
first  to  point  out  the  true  construction, 
which  requires  the  section  of  the  screw 
thread  to  be  a  rack  tooth,  although  it  need 
FIG.  247.  not  be  disputed  that  the  difference  would 

not  be  conspicuous,  nor  that  if  they  were  exactly  alike  they  would 

engage  and  transmit  rotation,  but  with  a  slight  fluctuation  in  the 

velocity  ratio. 

410.  Screw  and  Rack. — There  is  no  limit  to  the  increase  in  the  num- 
ber of  teeth  upon  the  worm  wheel,  and  if  it  be  made  infinite,  the 
wheel  becomes  a  rack,  which  if  cut  by  the  screw  itself,  will  be  iden- 
tical with  a  portion  of  an  ordinary  nut. 

Now  the  exterior  surface  of  the  screw  and  the  interior  surface  of  the 
nut  are  precisely  the  same  ;  and  this  affords  an  illustration  of  the 
extreme  case  in  deriving  the  conjugate  to  a  given  rack  tooth,  men- 
tioned in  (282),  for  if  we  split  the  nut  and  screw  longitudinally 
through  the  axis,  the  meridian  sections  will  correspond  to  the  two 
conjugate  racks  shown  in  Fig.  163,  being  exactly  converse  to  each 
other. 

Oblique  Screw  Gearing. 

411.  The  axis  of  the  screw  thus  far  has  been  supposed  to  lie  in  the 
plane  of  rotation  of  the  wheel.     This,  however,  is  not  essential,  for 
even  if  it  cross  that  plane  obliquely,  rotation  can  still  be  transmitted 
with  a  constant  velocity  ratio,  by  the  end  thrust  of  the  screw. 

And  in  this  new  relative  position  of  the  axes,  as  before,  the  screw 
may  have  two,  three,  or  any  number  of  threads,  and  thus  we  pass 
from  a  simple  endless  screw  to  the  disguised  forms  of  oblique  screw 
wheels. 


Principles  of  Mechanism,  p.  163. 


OBLIQUE   RACK   AKD   WHEEL. 


271 


The  fundamental  principle  remains  unchanged,  however  ;  the  screw 
is  still  a  rack  which  advances  hy  rotation,  and  the  first  step  in  the  con- 
struction is  to  determine  the  form  of  its  conjugate  tooth  with  refer- 
ence to  the  new  conditions. 

412.  The  Oblique  Eack  and  Wheel. — That  this  is  the  case,  may  be 
perhaps  most  readily  seen  by  first  considering  the  things  which  may 
be  accomplished  by  the  rack  and  wheel  alone. 

Ordinarily,  as  is  well  known,  the  rack  travels  in  the  plane  of  rota- 
tion of  the  wheel.  But  this,  again,  is  not  a  matter  of  necessity ; 
without  the  slightest  change  in  the  forms  of  the  teeth,  it  may  be  made 
to  travel  obliquely  across  that  plane,  the  velocity  ratio  remaining  ab- 
solutely constant,  although  its 
value  will  be  changed. 

This  is  illustrated  by  Fig. 
248  ;  if  we  suppose  a  rack  to  be 
made  by  cutting  teeth  across  a 
broad  rectangular  plate,  MN, 
indicated  by  the  dotted  lines,  it 
will  at  once  be  seen  that  it  can- 
not only  move  from  right  to 
left,  causing  the  wheel  to  turn 
as  usual,  but  is  also  free  to  slide  Mr 
in  the  direction  of  the  axis,  and 
that  it  may  receive  both  motions 
at  once. 

If  now  a  strip  be  cut  diagon- 
ally from  this  broad  rack,  and 
made  to  travel  by  guide  rollers, 
as  shown,  or  by  any  other  means, 
in  the  direction  of  that  diag- 
onal, the  effect  is  precisely  the  FIG.  248. 
same.  The  action  between  the  teeth  is  unchanged  ;  but  assigning 
a  definite  linear  velocity  to  the  rack  in  the  new  direction,  that  motion 
may  be  resolved  into  two  components,  one  lying  in  the  plane  of  ro- 
tation, the  other  perpendicular  to  it.  The  latter  does  not  affect  the 
wheel,  but  the  former  does,  and  causes  it  to  rotate  ;  and  the  linear 
velocity  of  the  pitch  circumference  is  equal  in  magnitude  to  this  effec- 
tive component. 

It  will  readily  be  seen  that  another  rack  may  be  cut  upon  the  back 
of  this  one,  with  teeth  perpendicular  to  its  sides,  which  may  engage 
with  another  wheel  in  the  ordinary  manner ;  and  thus  we  have  a  new 
means  of  transmitting  a  limited  rotation  with  a  constant  velocity 


272 


OBLIQUE   SCEEW  GEARING. 


ratio  between  two  axes  in  different  planes,  by  the  use  of  common 
spur  gearing  only. 

413.  The  resolution  above  mentioned  is  represented  in  Fig.  248  ; 
the  obliquity  of  the  rack's  travel,  when  assigned,  gives  the  direction 
of  the  resultant,  and  if  the  component  in  the  plane  of  rotation  be 
made  equal  to  the  pitch  arc,  the  magnitude  of  the  resultant  deter- 
mines the  distance  through  which  the  rack  will  advance  while  the 
wheel  turns  through  the  pitch  angle. 

Let  us  assume  this  distance  as  the  helical  pitch  in  constructing 
from  the  rack  an  oblique  single-threaded  worm,  as  shown  in  Fig.  249, 


\ 


K 


PIG.  249. 


A  being  its  pitch  cylinder,  B  that  of  the  wheel,  P  their  point  of  tan- 
gency.  Let  P  be  also  the  present  point  of  contact  between  a  thread 
and  a  tooth,  as  shown  at  P\  below;  when  cut  by  the  plane  LM,  normal 
to  the  axis  of  the  ivheel,  let  the  section  of  the  thread  have  the  form  of 
the  rack  tooth  in  the  preceding  figure,  and  that  of  the  wheel- tooth  be 
conjugate  to  it  as  before. 

As  the  screw  rotates  there  will  always  be  a  sectiornof  its  thread  sim- 
ilar to  this,  similarly  situated  with  regard  to  its  axis.     This  will 


Oy   THE 

^UNIVERSITY 

ALL  TRANSVERSE  SECTIONS   OF   WHEEL 

travel  along  with  uniform  speed,  as  indicated  by  the 
advancing  in  one  rotation  to  the  new  position,  0. 

Now  in  order  that  the  velocity  ratio  may  remain  strictly  constant, 
this  travelling  section  of  the  thread  must  be  always  acting  against  a 
tooth  outline  of  the  same  form.  Consequently,  in  every  transverse 
section  of  the  wheel,  the  teeth  will  be  bounded  ~by  similar  curves, 
although,  as  will  subsequently  appear,  they  will  not  necessarily  be  of 
uniform  height. 

414.  We  have  now  to  consider  the  twist  of  the  wheel  itself,  which 
depends  upon  the  same  principles  as  in  the  common  arrangement. 
Thus,  suppose  the  thread  and  the  tooth  in  contact  at  P,  to  be  grad- 
ually reduced  in  size;  they  will  ultimately  bedome  two  helical  lines, 
lying  one  upon  each  pitch  cylinder,  still  tangent  to  each  other,  and 
therefore  developing  upon  the  common  tangent  plane  into  the  same 
straight  line.     Thus,  if  the  pitch  of  the  screw-helix  be  given,  that  of 
the  other  is  found  as  in  Fig.  245,  and  is'-  the  same  for  all  the  helices 
of  the  teeth  of  the  wheel,  which,  if  cut  from  a  cylindrical  blank,  will 
be  a  simple  twisted  one,  as  in  Fig,  110,  with  all  its  transverse  sections 
alike.    • 

415.  Careful  consideration  of  the  action  will  show  that  the  surfaces 
generated  as  above  explained  are  precisely  such  as  the  screw  would 
cut  for  itself  under  the  assumed  conditions,  and   thus  confirm  the 
previous  statement  (401)  that  the  proper  forms  of  the  tooth  in  this, 
as  well  as  in  the  common  arrangement  of  screw  gearing  with  axes  in 
planes  mutually  perpendicular,  are  to  be  determined  by  the  principles 
which  apply  to  the  case  of  a  wheel  working  with  a  rack,  and  not  with 
another  wheel  of  any  finite  radius  whatsoever ;  notwithstanding  the 
fact  that  the  latter  construction  is  the  one  given  by  Prof.  Rankine.* 
We  would  not  be  understood  to  assert  that  two  oblique  screw-wheels 
will  absolutely  refuse  either  to  engage   or  to  work  with  each  other 
because  otherwise  fashioned,  but  that  the  method  of  construction 
here  set  forth  is  the  only  one  by  which  perfect  theoretical  precision 
can  be  attained. 

416.  Peculiar  Features  of  the  Action.— The  teeth  of  the  wheel  in 
Fig.  249  are  sections  by  successive  transverse  planes  through  R,  P,  0 ; 
their  conjugates,  R,  P',  0',  being  cut  from  the  screw  by  the  same 
planes. 

We  now  observe  that  one  rotation  of  the  worm  will  drive  the  wheel 
through  more  than  the  original  pitch  angle,  although  the  helical 
pitch  is  equal  to  the  diagonal  pitch  of  the  rack  in  the  preceding 

*  Machinery  and  Mill- Work,  p.  163. 


274 


OBLIQUE   SCREW   GEARISTG. 


figure.  In  that  case,  the  rack-tooth  was  always  acting  against  a  sur- 
face whose  elements  were  right  lines  perpendicular  to  the  plane  of 
rotation.  But  the  worm  acts  against  helices  which  cross  the  plane  of 
rotation  obliquely,  and  with  different  degrees  of  obliquity,  for  they  all 
have  the  same  pitch,  but  lie  at  different  distances  from  the  axis. 
Since,  however,  the  velocity  ratio  is  constant,  we  may  confine  our 
attention  to  the  helices  upon  the  pitch  cylinders,  and  study  their 
action  as  represented  in  the  development  of  these  surfaces  upon  the 
common  tangent  plane. 

Thus,  in  Fig.  249,  having  drawn  PK,  perpendicular  to  PO  and 
equal  to  the  circumference  of  the  pitch  cylinder  A,  OK  is  the  devel- 
oped helix  through  0,  and  PE  is  the  length  of  the  new  pitch  arc  for 
the  wheel.  If  then  a  complete  wheel  is  to  be  made,  the  proportions 
must  be  such  that  PJZis  an  aliquot  part  of  the  pitch  circumference. 

417.  And  all  this  accords  precisely  with  the  fundamental  principles 
relating  to  the  composition  and  resolution  of  motions.  In  Fig.  250 
CCbG  the  elements  of  the  pitch  cylinders  of  the  worm 
and  the  wheel  respectively,  inter- 
secting  at  P  ;  PO  the  pitch  of  the 
worm,  PK  perpendicular  to  PO, 
and  equal  to  its  pitch  circumfer- 
ence, and  OK  the  developed  helix, 
all  as  in  the  preceding  figure  ;  then 
PM  parallel  to  OK  is  the  developed 
helix  through  P,  which  point  we 
will  assume  as  before  to  be  the  pres- 
ent point  of  contact  between  a 
tooth  and  a  thread.  By  the  turn- 
ing of  the  worm,  the  point  P 
virtually  advances  uniformly  in  the 
direction  PD,  going  in  one  revolu- 
tion to  the  new  position  0,  while 
meantime  the  coincident  point  P 
of  the  wheel  must  move  in  the  di- 
rection PE.  Regarding  PM  as  a 
helix  of  the  wheel,  and  supposing  the  screw  to  be  pushed  endlong  in 
the  manner  of  a  rack,  we  observe  that  PO  may  be  resolved  into  the 
components  PM,  PE.  Of  these,  the  first  is  simply  the  sliding  compo- 
nent and  ineffective  ;  but  PE  is  the  one  which  must  represent  the  rota- 
tion of  the  wheel,  due  to  the  supposed  rectilinear  motion  of  the  screw. 
When,  however,  we  suppose  the  worm  to  have  a  motion  of  rotation 
only,  in  the  direction  shown  by  the  arrow  in  Fig.  249,  let  PK  repre- 


FIG.  250. 


ANALYSIS  OF  THE  ACTIOK.  275 

sent  the  linear  motion  of  its  driving  point  P,  which  acts  against  the 
helix  PM  of  the  wheel.  The  normal  and  tangential  components  of 
PK  are  respectively  PG  perpendicular  to  PM,  and  PJ  coincident 
with  it.  The  resultant  motion  of  the  point  P  of  the  wheel  must  be 
in  the  direction  PF,  perpendicular  to  (7(7,  and  of  such  magnitude,  PE, 
as  to  have  the  same  normal  component,  PG. 

The  same  method  mighfc  have  been  used  in  relation  to  the  recti- 
linear motion  PO  of  the  screw  when  used  as  a  rack,  and  with  concor- 
dant results,  since  PG  is  the  normal  component  of  that  motion  also. 

418.  In  all  the  foregoing  the  diameter  of  the  pitch  cylinder  of  the 
worm  was  assumed  at  pleasure,  which  it  may  be,  subject  to  the  condi- 
tion above  pointed  out,  that  PE  must  be  an  aliquot  part  of  the  pitch 
circumference  of  the  wheel. 

It  is  perfectly  possible,  then,  that  a  double  as  well  as  a  single  thread 
might  be  made  upon  a  pitch  cylinder  of  given  size.  Supposing  this 
to  be  desired,  and  also  that  the  subdivision  of  the  given  wheel  should 
remain  unchanged,  it  is  then  evident  that  in  Fig.  250  we  must  double 
PE,  the  pitch  arc,  and  not  PO,  the  helical  pitch  of  the  screw,  which 
will  now  become  PN,  determined  by  producing  KFto  cut  DD  in  N. 
We  thus  determine  new  helices,  KN,  PL,  with  reference  to  which  the 
motion  PK  is  to  be  resolved  as  before,  PH  being  the  normal  or  effect- 
ive, jP/the  tangential  or  sliding  component. 

419.  The  diagram,  Fig.  250,  is  drawn  without  regard  to  practical 
proportions,  the  conditions  being  selected  with  a  view  to  illustrating 
one  particular  in  which  oblique  screw  gearing  may,  in  some  cases, 
differ  from  the  ordinary  arrangement.     We  have  seen  that  when  the 
axes  lie  in  planes  perpendicular  to  each  other,  the  helices  on  the  worm 
and  the  wheel,  whatever  the  number  of  threads  and  teeth,  are  either 
both  right-handed  or  both  left-handed  (403). 

Now  in  Fig.  250,  both  KO  and  KN  will  form  right-handed  helices 
upon  the  pitch  cylinder  of  the  screw  ;  but  on  wrapping  the  tangent 
plane  down  upon  the  wheel,  KN  will  become  a  right-handed,  KO  a 
left-handed,  helix.  Evidently  there  is  an  intermediate  position  in 
which  the  developed  worm-helix  will  be  parallel  to  CC,  and  will, 
therefore  become  a  rectilinear  element  of  the  wheel's  pitch  cylinder. 
In  that  case  the  screw  will  work  with  a  common  spur-wheel ;  and  the 
proportions  which  must  obtain  in  order  to  secure  this  result  are  shown 
in  Fig.  251.  If  the  pitch  arc,  PE,  and  the  obliquity,  CPD,  are  both 
assigned,  the  pitch  and  circumference  of  the  screw  are  determined 
by  drawing  through-  E  a  perpendicular  to  PE,  cutting  DD  at  0,  and 
^^perpendicular  to  DD  at  K.  If  the  pitch  arc,  PE,  and  the  circum- 
ference PK  are  assigned,  we  first  describe  an  arc  about  P  with  radius 


276  CLOSE-PITTING   OBLIQUE   SCREW. 

PE\  draw  KE  tangent  to  it,  and  also  PD  perpendicular  to  PK. 
Then  KE  produced  cuts  PD  in  0,  determining  the  pitch  PO  and 
the  obliquity  OPE. 

420.  Close-fitting  Oblique  Tangent-screw.— The 
oblique  worm  may  be  used  to  cut  its  own  wheel, 
just  as  in  the  common   arrangement ;  and  the 
wheel  itself,  instead  of  being  cut  from  a  plain 
cylindrical  blank,  may,  for  the  sake  of  a  neater 
appearance,  conform  somewhat  to  the  curvature 
of  the  screw,  and  be  terminated  by  conical  frusta 
instead  of  transverse  planes.     The  form  of  the 
blank  is  fixed  by  very  simple  considerations ; 
regarding  the  screw  as  a  thread  wrapped  around 
a  cylindrical  core,  we  perceive  that  the  tops  of 
the  wheel-teeth  must  just  go  clear  of  that  core 
during  the  rotation.     Now  imagine  the  whole  screw  to  revolve  about 
the  axis  of  the  wheel ;  then  the  element  of  the  core  which  lies  nearest 
to  that  axis  will  generate  a  hyperboloid,  and   if  the  blank  for  the 
wheel  be  turned  to  this  exact  size,  the  tops  of  the  teeth  will  just 
touch  the  core.     It  should  therefore  be  made  a  little  smaller. 

It  will  be  seen  that  the  screw  still  travels  obliquely  across  the  plane  of 
rotation,  so  that  the  transverse  sections  of  the  teeth  are  all  of  the  same 
form,  contact  existing  between  each  thread  and  its  tooth  at  a  single 
point  only  ;  whence  the  only  advantage  gained  by  the  adoption  of  this 
form  of  blank  lies  in  the  fact  that  the  teeth  become  higher  as  they 
recede  from  the  mid-plane,  and,  therefore,  continue  longer  in  action. 
421.  Oblique  Screw  and  Rack. — Further,  it  will  be  seen  that  the  di- 
ameter of  the  oblique  worm-wheel  may  be  increased  at  pleasure  until 
it  eventually  loses  its  curvature,  and  assumes  the  form  of  a  rack. 
Under  these  circumstances  Prof.  Rankine's  instructions,*  if  we  inter- 
pret them  correctly,  are  to  the  effect  that  the  normal  section  of  the 
worm-thread  should  be  that  of  the  tooth  of  a  wheel  working  with  a 
rack,  the  tooth-outline  of  the  latter  to  be  adopted  as  the  normal  sec- 
tion of  the  rack  to  work  with  this  worm  ;  and  he  gives  specific  direc- 
tions for  finding  the  radius  of  that  wheel  in  any  given  case. 

Without  disputing  that  the  thread  and  tooth  thus  formed  will 
work,  we  would  remark  that  it  is  not  the  only  nor  yet  the  best 
manner  of  forming  them.  The  tooth-surfaces  of  the  rack  must  be 
made  up  of  parallel  rectilinear  elements,  and  since  in  the  case  of  a 
rack  and  wheel  the  teeth  have  but  a  single  point  of  contact,  the 

*  Machinery  and  Mill-Work,  p.  290. 


OBLIQUE   SCREW   AND   RACK.  277 

tooth-surface  of  the  rack  will,  in  Prof.  Rankine's  construction,  touch 
each  other  in  that  point  only.  Which  would  hold  true  at  the  limit 
when  the  obliquity  vanishes  ;  whereas  it  is  perfectly  patent  that  the 
rack  may  then  be  a  part  of  a  nut,  and  its  whole  surface  in  contact 
with  that  of  the  screw-thread,  whatever  the  form  of  the  latter.  This 
superficial  contact  is  not  attainable  when  the  rack  travels  obliquely, 
it  is  true  ;  but  line-contact  between  the  acting  surfaces  may  be  secured 
with  a  screw-thread 'of  any  reasonable  form,  as  will  appear  from  the 
following  considerations. 

422.  Taking  a  common  triangular-threaded  screw  in  illustration, 
we  have,  in  Fig.  252,  a  meridian  section  on  the  left,  and  an  outside 
view  on  the  right,  the  axis  being  parallel  to  the  paper.    The. apparent 
contour  of  the  completed  screw 

is  the  trace  of  its  projecting 
cylinder,  whose  elements  are 
perpendicular  to  the  paper,  and 
tangent  to  the  surface.  If  then 
a  rack  be  made,  as  shown  in 
section  below,  its  teeth  bounded 

by  the  same  outlines  and  their  elements  also  perpendicular  to  the 
paper,  these  teeth  will  always  touch  the  screw  along  the  line  of  its 
visible  contour,  whether  the  rack  be  made  to  travel  endlong  by  turn- 
ing the  screw,  or  to  slide  transversely,  or  both.  And  if  these  actions 
occur  simultaneously,  we  have  the  oblique  motion  desired. 

Line-contact  is  thus  obtained,  and  the  normal  section  of  the  rack- 
tooth  is  determined  by  merely  drawing  the  screw,  in  its  simplest 
position.  But  this  is  not  the  only  form  which  will  effect  the  same 
result ;  for  in  whatever  direction  we  chose  to  look  at  the  screw,  if  the 
space  between  the  threads  be  visible  at  all,  the  line  of  apparent  con- 
tour will  give  the  normal  section  of  a  tooth  for  a  rack  which  will 
work  in  line-contact  with  it,  cither  longitudinally  or  obliquely. 

423.  Among  so  many  forms,  the  selection  of  the  best  may  be  safely 
left  to  the  screw  itself,  which  will  assert  its  preference  if  given  the 
opportunity.     This  may  be  afforded  by  allowing  it  to  cut  its  own 
rack,  the  latter  being  moved  by  independent  means  m  the  required 
direction,  and  at  the  same  speed  as  that  which  it  is  eventually  to  re- 
ceive from  the  screw. 

It  is  clear  that  the  amount  of  metal  removed  will  be  the  least  possi- 
ble ;  whence  the  deduction  that  the  normal  section  of  the  rack-tooth 
will  be  the  visible  contour  of  the  space  between  the  screw-threads 
when  viewed  from  the  direction  which  will  give  the  greatest  apparent 
breadth  to  that  space. 


278 


MINIMUM  AMOUNT  OF  SLIDING. 


What  that  precise  direction  is,  may  depend  somewhat  upon  the 
meridian  section  of  the  screw,  but  probably  will  not  vary  appreciably 
from  that  of  the  tangent  to  the  helix  of  mean  obliquity. 

It  will  be  seen  by  reference  to  Fig.  251,  that  if  the  pitch,  diameter, 
and  inclination  of  the  screw  to  the  line  of  travel  be  properly  propor- 
tioned, the  rack  may  be  made  of  the  usual  form,  that  is,  with  the 
teeth  cut  transversely  across  its  face.  But  whatever  the  arrangement, 
proportions,  or  form,  the  kinematic  action  consists  of  sliding  contact 
pure  and  simple ;  there  is  not,  as  has  been  sometimes  erroneously 
stated,  the  slightest  admixture  of  rolling  contact,  or  of  anything  even 

distantly  resembling  it. 

424.  To  Secure  the  Least 
Amount  of  Sliding, — Although 
the  velocity  ratio  depends  upon 
the  pitch  and  not  upon  the 
size  of  the  screw,  yet  it  is  evi- 
dent that  for  a  given  pair  of 
axes,  and  a  given  velocity  ratio, 
there  must  be  some  definite  ratio 
between  the  diameters  of  the 
pitch  cylinders,  which  will  in- 
volve less  sliding  than  any  other. 
"What  this  proportion  is,  may 
be  thus  deduced.  If  upon  the 
given  axes  we  construct  a  pair 
of  rolling  hyperboloids  with 
the  assigned  velocity  ratio,  these 
surfaces  will  work  in  contact 
with  no  sliding  other  than  that 
along  the  common  element. 
This  element  passes  through 
the  common  point  of  the  gorge 
circles,  and  lies  in  a  plane  tan- 
gent at  that  point  not  only  to  both  hyperboloids,  but  to  their  inscribed 
tangent  cylinders  of  which  the  gorge  circles  are  the  bases. 

Now,  if  these  be  taken  as  the  pitch  cylinders  of  the  screw  wheels, 
and  the  common  element  as  the  developed  helix,  the  amount  of  slid- 
ing will  be  the  same  as  that  between  the  two  hyperboloids  when  rota- 
tating  w'ith  the  given  velocity  ratio. 

And  this  amount  is  then  a  minimum.  In  Fig.  253,  let  PI  repre- 
sent the  motion  of  the  point  P  of  the  inclined  surface ;  if  the  ver- 
tical one  be  driven  by  it,  we  find  in  the  usual  manner  the  motion  of 


FIGS.  253  and  254. 


•v- 


KESEMBLANCE   TO    SKEW   WHEELS.  279 

the  coincident  point  to  be  PL,  and  that  the  sliding  is  represented  by 
NM  on  the  line  of  contact.  PI  remaining  unchanged,  let  the  verti- 
cal hyperboloid  be  compelled  to  revolve,  by  independent  means,  faster 
than  the  other  should  drive  it ;  and  let  PK  be  the  new  velocity  of 
its  point  P.  The  sliding  component  along  PB  now  becomes  PO, 
which  is  greater  than  PN ';  and  there  will  be  also  a  sliding  perpen- 
dicular to  the  common  element,  represented  by  GH.  A  similar 
result  would  follow,  were  PI  to  bo  increased  whilo  PL  remained  the 
same  ;  whence  it  appears  that  the  sliding  is  least  when  the  velocity 
ratio  is  that  for  which  the  hyperboloids  were  constructed ;  and  this 
holds  true  of  the  tangent  cylinders,  the  obliquity  of  the  helices  being 
determined  as  above. 

425.  The  sliding  action  represented  by  GH  cannot  actually  occur 
when  teeth  are  used  ;  but  the  linear  motions  PI  and  PK  can  still  be 
retained  in  the  case  of  screw  gearing,  as  shown  in  Fig.  254,  by  mak- 
ing the  developed  helix  PB  parallel  to  KI\  all  the  sliding  must 
necessarily  be  in  this  direction,  and   its  amount  OM  is,  of  course, 
greater  than  before.     This  proceeding,  it  will  be  observed,  is  merely 
the  converse  of  that  explained  in  connection  with  Fig.  250,  the  results 
being  exactly  concordant. 

If  the  axes  lie  in  planes  perpendicular  to  each  other,  the  hyperbo- 
loids have  two  common  elements,  either  of  which  may  be  taken  as  the 
developed  helix,  according  to  the  directional  relation  desired ;  both 
wheels  will  be  right-handed  in  one  case,  and  left-handed  in  the  other. 

426.  Resemblance  to  Skew  Wheels. — The  appearance  of  screw  gear- 
ing thus  constructed  is  quite  unlike  that  of  the  familiar  worm  and 
wheel.     In  that  combination  the  wheel  cannot  move  the  screw,  nor 
is  it  usually  desirable  that  it  should,  since  in  most  ea^es  the  latter  is 
required  not  only  to  turn  the  wheel,  but  to  hold  it  in  any  given  posi- 
tion.    Eut  in  the  arrangement  now  under  consideration  either  wheel 
may  be  used  as  the  driver  ;  it  is  quite  apparent  that  a  single-threaded 
worm  made  in  the  manner  above  described  could  not  work  unless  it 
were  of  great  length  and  its  axis  nearly  parallel  to  that  of  the  wheel, 
but  that  this  construction  is  adapted  only  for  the  teeth  of  screw  wheel- 
work  properly  so  called  ;  and  equally  evident  that  for  this  it  is  the 
best. 

The  mid-planes  of  these  wheels  are  the  gorge  circles  of  the  tangent 
hyperboloids ;  the  pitch  helices  are  tangent  to  the  elements  of  those 
surfaces ;  and  from  the  considerations  presented  in  (420),  it  will  be 
seen  that  both  blanks  may  be  made  of  hyperboloidal  outline.  From 
all  which,  there  results  a  resemblance  between  these  and  skew  wheels 
sufficiently  close  to  be  misleading. 


280 


HOUR-GLASS   WORM   GEAR. 


This  resemblance,  however,  is  superficial ;  the  teeth  are  made  up 
of  cylindrical  helices,  touch  each  other  in  a  single  point  only,  can  be 
extended  only  to  a  short  distance  either  way  without  ceasing  to 
engage,  and  their  acting  sections  are  respectively  those  of  a  spur  wheel 
and  a  rack. 

And  the  one  whose  section  corresponds  to  that  of  the  rack-tooth 
may  be  made  of  any  length,  and  if  desired  may  be  used  as  a  rack, 
driving  the  other  wheel  not  by  rotation,  but  by  moving  longitudinally, 
whatever  the  inclination  of  the  axes  or  the  number  of  threads. 

Hindley's  Screw,  or  Hour-glass   Worm. 

427.  A  Worm  may  be  so  shaped  as  to  conform  to  the  curvature  of 

IO  the   wheel,  assuming  a  figure 

somewhat  like  that  of  an  hour- 
glass ;  in  Fig.  255,  the  complete 
worm  is  represented  on  the  left, 
and  on'  the  right  is  given  a  sec- 
tion through  its  axis  by  a  plane 
perpendicular  to  that  of  the 
wheel.  The  surface  of  the 
thread  is  of  a  complicated  and 
peculiar  nature,  but  practically 
it  is  very  easily  made,  thus  :  a 
tool,  with  a  cutting  point  of  the 
contour  of  the  wheel's  tooth,  is 
so  clamped  to  a  disc  that  its 
upper  surface  lies  in  the  meridian  plane  of  the  worm,  and  both  the 
disc  and  the  worm  blank  are  driven  by  intermediate  gearing  at  their 
proper  relative  velocities. 

In  this  manner  the  screw  was  made  by  Hindley,  who  first  intro- 
duced it.  For  some  reason  it  has  never  come  into  general  use, 
although  in  smoothness  and  steadiness  of  action  it  would  appear  to 
be  superior  to  the  common  form  of  tangent-screw. 

The  worm  itself  may  now  be  formed  into  a  cutter,  and  made  to 
finish  its  own  wheel  in  the  usual  manner  ;  which  is  the  method 
adopted  by  Messrs.  Clem  &  Morse  of  Philadelphia,  Penn.,  who  have 
recently  constructed  a  very  elegant  engine  specially  designed  for  cut- 
ting this  description  of  screw  gearing. 

428.  The  outline  of  the  pitch  surface  of  this  worm  is  an  arc  of  the 
pitch  circle  of  the  wheel.     Upon  this  suface  the  helix  EFP,  in  Fig. 
256,  is  traced  by  a  point  which  moves  about  the  centre  C  through  the 
arc  EP,  while  the  worm  makes  one  revolution,  both  motions  being 


FIG.  255. 


FORM   OF   PITCH   SURFACE. 


281 


uniform  in  velocity.     The  longitudinal  advance,   UV,  is  not  equal  to 
EP  ;  it  is  not  uniform,  nor  is  it  the  same  for  successive  convolutions. 

The  projection  of  this  helix  upon  a  plane,  OC,  perpendicular  to  DD, 
as  shown  at  the  right,  is  of  a  spiral  form,  resembling  that  of  the  hy- 
perboloidal  helix  in  Fig.  243  ;  to  which  this  curve  is  very  similar. 
Other  points  in  the  outline  of  the  tooth,  either  within  or  without  the 
pitch  circle,  also  dc-  N 

scribe  curves  of  the 
same  nature,  lying 
upon  surfaces  whose 
meridian  outlines  are 
circular  arcs ;  these 
curves  are  easily 
drawn,  and  their 
envelope  is  the  visi- 
ble contour  of  the 
worm. 

The  form  of  the 
threads  is,  abstractly, 
arbitrary ;  their  me- 
ridian sections  are  FIG.  256. 
exactly  converse  to  those  of  the  wheel-teeth,  the  spaces  between 
which,  as  seen  in  Fig.  255,  they  fit  and  fill  entirely,  throughout  the 
action.  Thus  there  is  no  relative  motion  in  the  manner  of  a  rack  or 
otherwise,  but  the  worm  is  locked  into  the  wheel,  and  can  move  only 
by  revolving  about  one  axis  or  the  other.  A  good  and  simple  practi- 
cal form  for  the  teeth  of  the  wheel  is  that  given  in  Fig.  255,  the  sides 
being  straight  sloping  lines,  equally  inclined  to  the  radius  of  symme- 
try ;  the  amount  of  inclination  is  fixed  by  the  consideration  that,  in 
the  extreme  position  of  the  action,  the  outer  side  should  lie  in  a  line, 
AB,  perpendicular  to  DD,  in  order  that  the  worm  may  be  readily 
disengaged  from  the  wheel. 

429.  The  Action  Confined  to  the  Mid-plane  of  the  Wheel. — The  ad- 
vantage of  the  hour-glass  worm  lies  in  the  fact  just  stated,  that  the 
whole  side  of  every  thread,  in  the  meridian  plane,  is  always  in  con- 
tact with  the  adjacent  tooth. 

It  has  been  asserted  that  the  teeth  of  the  wheel,  when  cut  by  the 
worm,  touch  the  threads  of  the  latter  at  all  points ;  *  or,  in  other 
words,  that  the  whole  surfaces  are  in  contact. 

Either  one  of  two  considerations  is  sufficient  to  show  that  this  is 


"Mechanics  "  for  January  14,  1882. 


282  THE   ACTION   CONFINED  TO   ONE   PLANE. 

impossible.  Let  any  side  plane  be  passed,  parallel  to  the  axis  of  the 
worm  and  perpendicular  to  that  of  the  wheel  ;  then,  in  the  first  place, 
the  sections  of  the  successive  threads  will  not  be  similar ;  and,  in  the 
second  place,  the  section  of  the  pitch  surface  of  the  worm  will  not  be 
a  circle,  whereas  the  teeth  of  the  wheel  travel  in  circular  paths. 
Therefore,  such  superficial  contact  is  attainable  only  when  the  radii 
of  these  paths  become  infinite,  in  which  event  the  pitch  surface  of 
the  worm  becomes  a  cylinder,  the  worm  itself  a  common  screw,  and 
the  wheel  a  portion  of  a  nut ;  but,  in  general,  the  action  is  confined 
to  the  central  plane  of  the  wheel. 

430.  The  Pitch  Surface  of  the  Wheel  may  to  some  extent  conform  to 
the  curvature  of  the  worm,  as.  shown  in  section  in  the  right-hand 
part  of  Fig.  256.  Any  side  plane,  L M,  parallel,  to  the  mid-plane,  NR, 
cuts  the  base,  Iff,  of  the  worm's  pitch  surface  in  a  point  L,  which 
when  revolved  about  the  axis  of  the  wheel  into  the  gorge  plane  00, 
takes  the  position  L',  thus  giving  a  point  in  the  contour  L P'S. 

By  a  similar  process,  reducing  the  radius  VP  of  the  gorge  of  the 
worm,  and  increasing  the  radius  CP,  according  to  the  height  of  the 
tooth,  we  may  determine  the  outline  of  the  blank.  This  conforma- 
tion of  the  wheel  does  not,  it  must  be  noted,  secure  any  additional 
contact,  or  in  any  way  affect  the  action,  but  merely  gives  a  neater 
finish  than  if  the  blank  were  made  a  plain  cylinder. 

431.  The  Teeth  of  the  Wheel  are  automatically  shaped  by  the  cutting 
worm  with  perfect  ease;  but  the  accurate  delineation  of  the  form 
thus  determined  is  a  rather  complicated  and  tedious  matter.     Still,  if 
necessary,  it  can  be  accomplished  in  the  following  manner. 

When  the  outline  of  the  blank  has  been,  found,  as  above,  it  will  be 
seen  that  any  transverse  plane  will  cut  from  it  a  circle,  which  will 
limit  the  height  of  the  teeth.  The  same  plane  will  cut  from  the 
threads  of  the  screw  a  series  of  sections  of  varying  form,  each  lying 
a  little  in  advance  of  the  meridian  section  of  the  same  thread.  The 
sections  of  the  wheel-teeth  by  the  same  plane  must  lie  in  the  spaces 
between  those  of  the  screw-threads  ;  and  since  each  revolution  of  the 
worm  turns  the  wheel  one  tooth  ahead,  it  follows  that  each  tooth- 
section  must  take  in  succession  all  the  different  positions  thus  deter- 
mined. We  may,  then,  proceed  thus :  lay  a  piece  of  tracing  paper 
over  the  drawing,  and  secure  it  by  a  pin  fixed  at  the  centre  of  the 
wheel.  Upon  this,  trace  the  sections  of  two  consecutive  threads,  and 
also  the  circular  arc  bounding  the  top  of  the  tooth  in  this  plane. 
Then  rotate  the  paper  about  the  central  pin  through  the  pitch  angle, 
again  trace  the  sections  of  the  two  adjacent  threads,  and  so  on  until 
all  have  been  traced  ;  the  clear  space  within  the  lines  thus  drawn 


WORM  WHEEL   WITH   ROLLERS. 


283 


will  be  the  required  section  of  the  tooth  by  the  given  plane.  Abso- 
lute certainty  of  definition  will  require  the  screw  to  be  placed  in  sev- 
eral positions,  cut  in  each  by  the  same  plane,  and  the  above  process  to 
be  repeated  for  each,  using  the  same  tracing  throughout ;  because  the 
sections  of  these  peculiar  screw-threads  in  the  different  phases  of  rota- 
tion are  dissimilar.  And  finally,  this  whole  operation  must  also  be 
repeated  with  several  different  planes,  the  number  depending  upon 
the  degree  of  accuracy  aimed  at :  all  of  which  we  leave  the  reader  to 
execute  at  leisure. 

432.  Multiple-threaded  Hour-glass  Worm. — Though  we  are  not  aware 
that  this  form  of  worm  has  ever  been  made  with  two  or  more  threads, 
there  seems  to  be  no  abstract  objection  to  increasing  the  number. 
Also  it   is  clearly  possible,  by  making  the  screw  pitch  sufficiently 
large,  to  use  the  wheel 

as  the  driver :  in  which 
case  the  teeth  might  pre- 
ferably be  made  in  the 
form  of  turned  pins,  set 
into  the  periphery  of  a 
cylinder  as  in  face  gear- 
ing. Evidently  a  greater 
number  of  these  pins,  or 
teeth,  Avould  be  simul- 
taneously engaged  than 
in  the  common  form  of 
screw  wheels,  which 
would  certainly  tend  to 
increase  the  steadiness  of 
the  motion ;  but  the 
whole  action  is  confined, 
as  before,  to  the  meridian 
plane  of  the  worm  per- 
pendicular to  the  axis  of 
the  wheel. 

433.  Rollers    Substitu- 
ted for  Teeth. — With  the  purpose  of  reducing  the  sliding  friction  as 
much  as  possible,  the  singular  device  illustrated  in  Fig.  257  has  been 
proposed. 

The  meridian  section  of  the  screw-thread  is  bounded  by  right  lines 
at  right  angles  to  each  other,  and  the  worm  is  made  long  enough  to 
embrace  more  than  one  quarter  of  the  circumference  of  a  wheel,  the 
number  of  whose  teeth  must  be  a  multiple  of  four  ;  for  this  wheel  is 


FIG.  257. 


284: 


FACE   GEARING. 


substituted  a  frame  carrying  rollers,  arranged  in  the  manner  shown 
in  the  figure. 

The  object  of  diminishing  the  friction  is  certainly  accomplished  ; 
but,  on  the  other  hand,  only  one  thread  of  the  worm  is  in  action  at 
once,  during  the  greater  part  of  the  time.  And  that  one  has  at  no 
instant  more  than  a  single  driving  point ;  for,  since  the  warped  sur- 
face of  the  screw  cannot  be  placed  in  right-line  contact  with  a  surface 
of  single  curvature,  these  rollers  cannot  be  made  cylindrical,  as  they 
have  sometimes  been  represented  ;  but  their  contours  must  be  slightly 
convex  curves  tangent  to  the  meridian  sections  of  the  threads. 

Taking  these  drawbacks  into  consideration,  it  would  appear  that 
this  arrangement  is,  from  a  practical  point  of  view,  more  curious 
than  useful ;  although  it  might  serve  a  purpose  in  very  light-running 
mechanism  intended  rather  for  the  modification  of  motion  than  for 
the  transmission  of  power. 

The  Teeth  of  Face-Wheels. 

434.  Let  two  Wheels,  exactly  alike,  whose  teeth  are  cylindrical  pins 
fixed  in  the  faces  of  circular  discs,  be  so 
placed  that  each  axis  lies  in  a  plane  perpen- 
dicular to  the  other,  at  a  distance  from  it 
equal  to  the  diameter  of  the  pins,  as  in  Fig. 
258. 

Under  these  circumstances,  the  angle 
FDE  will  always  be  equal  to  the  angle 
HCG,  as  long  as  the  pin  E  of  the  driver  A 
is  in  contact  with  the  pin  G  of  the  follower 
B  :  the  velocity  ratio  is,  therefore,  perfectly 
constant. 

The  length  of  the  pin  E  must  be  such 
that  the  next  pin  /  of  the  other  wheel  shall 
not  catch  upon  its  end  in  going  into  gear. 
And  it  will  also  be  noted  that  although  the 
pin  7",  at  the  instant  when  the  next  pin  K 
of  the  driver  begins  to  act  upon  it,  may  also 
touch  the  pin  E  upon  the  back,  it  cannot 
continue  to  do  so.  It  is  not  possible,  there- 
fore, even  theoretically  to  secure  entire  free- 
FIG.  ass.  dom  from  backlash. 

The  maximum  length  of  one  pin  having  been  ascertained,  it  is  of 
course  the  same  for  all.  And  it  is  next  to  be  observed  that  if  the 
number  be  increased  this  length  must  be  diminished  ;  also,  that  there 


FACE   GEARING. 


285 


will  in  every  case  be  a  limit  beyond  which  the  number  cannot  be  in- 
creased without  at  the  same  time  diminishing  the  diameter,  and  in 
consequence  the  distance  between  the  axes. 

Ultimately,  then,  the  axes  will  intersect  at  right  angles,  and  the 
pins  will  become  consecutive  points  in  the  circumferences  of  two 
equal  circles  rolling  together  like  the  bases  of  the  pitch  cones  of  a 
pair  of  mitre-wheels.  In  other  words,  there  are  no  pitch  surfaces, 
these  degenerating  into  lines,  and  the  elementary  teeth  into  points. 

435.  Equal  Wheels  with.  Axes  at  Right  Angles.— The  preceding  is 
the  simplest  form  of  face  gearing,  but  it  cannot  be  used  if  the  axes 
intersect,  as  is  often  necessary.  When  they  do,  however,  the  teeth 
of  one  wheel  may  still  be  made  cylindrical :  ( 

those  of  the  other  being  solids  of  revolution, 
whose  outlines  may  be  determined  as  follows. 

In  Fig.  259,  let  the  two  wheels  be  of 
equal  size,  their  axes  perpendicular  to  each 
other.  Lot  E  be  a  pin  of  no  sensible  di- 
ameter, fixed  in  the  wheel  A,  and  F  a  simi- 
lar one  fixed  in  the  wheel  B,  the  distance 
between  the  two  being  arbitrary.  Let  the 
wheels  turn  through  equal  angles  as  indica- 
ted by  the  arrows  ;  the  pins  will  come  into 
the  new  positions  G,  H,  and  /,  K,  and  in 
the  meantime  their  common  perpendicular 
will  continually  change  both  in  position  and 
in  magnitude. 

But  it  is  always  easily  determined ;  and 
it  is  evident  that  if  we  now  make  F  the 
axis  of  a  surface  of  revolution,  the  radius  of 
each  transverse  section  being  this  common 
perpendicular,  we  shall  have  the  form  of  a 
pin  or  tooth  for  B,  which  will  work  with 
the  pin  E  of  no  sensible  diameter,  the  velocity  ratio  being  constant 
throughout. 

This  process  is  more  fully  illustrated  in  Fig.  260,  in  which  portions 
of  the  pitch  circles  only  are  shown.  The  action  is  readily  traced  : 
the  curve  El  will  be  generated  while  the  pins  ^and  F  traverse  the 
equal  arcs  EP,  FO,  and  the  curve  LG,  while  they  move  through 
the  arcs  PG,  OR,  which  are  also  equal  to  each  other  ;  the  ordinates 
x,  z,  being  found  by  placing  the  pins  in  the  intermediate  positions 
shown,  any  required  number  may  be  determined,  in  a  similar 
manner. 


FIG.  £ 


286 


FACE   GEARING. 


436.  Upon  assuming  a  sensible  diameter  for  the  pin  of  the  upper 
wheel,  we  proceed  to  derive  the  outline  of  the  working  tooth  of  the 
other,  precisely  as  was  done  in  the  case  of  pin  gearing,  as  shown  on 


the  right  in  the  figure ;  drawing  a  curve  at  a  constant  normal  dis- 
tance from  LG,  equal  to  the  radius  of  the  pin. 

Since  on  the  equal  circles  AA,  BB,  we  have  the  arcs  EP  =  FO  = 
F'O',  and  PG  =  OR  =  O'E',  it  follows  that  the  arcs  EF1,  P0\  GR, 
etc.,  are  also  equal  to  each  other.  Whence,  EP  being  equal  to  PG, 
we  find  that  EH  is  greater  than  GK\  similarly  x  is  greater  than  z, 
and,  in  general,  all  the  ordinates  of  El  are  greater  than  the  corre- 
sponding ordinates  of  GL,  the  difference  diminishing  as  we  descend, 
until  finally  IM  is  equal  to  LN. 

Consequently,  the  working  tooth  for  the  lower  wheel  must  be  de- 


Fio.  261. 

rived  from  the  curve  GL  and  not  from  El ;  and  in  order  to  secure 
receding  instead  of  approaching  action,  the  cylindrical  pins  must  be 
given  to  the  driver,  and  not  to  the  follower  as  in  pin  gearing. 

437.  Unequal  Wheels  with  Axes  at  Right  Angles.— In  Fig.  261,  let  the 
cylindrical  pins  be  given  to  the  smaller  wheel,  of  which  AP'A  is  the 
pitch  circle.  Assuming  any  point  E  upon  this  circumference,  draw 


FACE   GEARIKG. 


287 


through  it  a  parallel  to  CD,  cutting  the  larger  pitch  circle  BPB  in  F. 
Then  the  arc  EP'  is  greater  than  the  arc  FP  ;  therefore,  if  we  make 
FO  equal  to  EP',  a  parallel  to  CD  through  0  will  cut  the  tangent  at 
P'  in  sorgo  point  Ton  the  right  of  P'  ;  and  IM  will  be  the  lowest 
ordinate  of  the  curve  El,  which,  as  in  the  preceding  figure,  will  be 
generated  while  a  pin  at  E  traverses  the  arc  EP'  and  one  at  F  moves 
to  0.  Also  P' Twill  be  equal  to  LN9  the  lowest,  and  evidently  the 
least,  ordinate  of  the  curve  LG  generated  while  the  pins  traverse  the 
equal  arcs  OR  and  P'G.  The  whole  process,  including  the  determi- 
nation of  the  intermediate  ordinates,  is  the  same  as  before' ;  but  since 
the  lease  of  the  two  elementary  surfaces  must  necessarily  be  employed, 
we  perceive  that  the  working  tooth  in  this  case  must  be  derived  from 
El  instead  of  LG,  and  also  that  the  cylindrical  pins  must  be  given 
to  the  follower,  as  in  pin  gearing,  and  not  to  the  driver,  in  order  that 
the  action  may  be  receding. 

438.  As  the  diameter  cf  the  driver  becomes  greater  its  curvature 
becomes  less,  and  at  the  limit  will  disappear ;  the  combination  then 
becoming  identical  with  that  of  a  rack  driving  a  pin-wheel.     The 
curves  El  and  GL  will  in  that  case  evidently  be  similar  and  equal, 
each  being  the  cycloid  of  which  AA  is  the  generating  circle. 

There  is,  in  fact,  a  close  analogy  between  pin-gearing  and  face- 
gearing  ;  for  if  in  the  latter  we  suppose  the  axes  to  be  parallel,  the 
teeth  of  one  wheel  will  project  radially  from  the  curved  surface  of  a 
cylinder,  and  their  contours  will  be  precisely  like  those  of  a  spur- 
wheel  working  with  the  round  staves  of  a  trundle,  or  lantern  pinion. 

439.  But  the  cylindrical  pins  may  be  given  to  the  larger  of  the  two 
unequal    wheels.       The 

case  is  then  very  nearly 
the  converse  of  the  pre- 

:  the  rotations 

the  same  di- 
rection, the  pointed 
tooth,  as  seen  in  Fig. 
262,  appears  on  the  op- 
posite side  of  the  plane 
of  the  axes,  and  if  the 
action  is  to  be  receding, 
the  larger  wheel,  with  the  cylindrical  pins,  must  drive. 

The  greater  the  difference  between  the  diameters  of  the  pitch 
circles,  the  shorter  will  be  the  teeth  of  the  follower  for  a  given  arc  of 
action.  And,  in  consequence  of  this,  the  size  of  the  driver  cannot 
be  indefinitely  increased,  since  upon  reaching  the  limit,  its  pins 


A 


ceding  one 
being  in 


FACE   GEARING. 


would  move  in  a  plane  parallel  to  the  face  of  the  follower,  and  the 
tooth  of  the  latter  would  reduce  to  a  ssries  of  concentric  circles  lying 
in  the  same  plane. 

440.  Unequal  Wheels  with  Axes  in  Different  Planes. — Precisely  the 
same  method  of  construction  is  applicable  also  when  each  axis  lies  in 
a  plane  perpendicular  to  the  other,  although  they  do  not  meet,  and 
whether  the  wheels  be  equal  or  unequal  in  size.  Thus  in  Fig.  263, 

c 


K 


supposing  the  cylindrical  pins  to  be  assigned  to  the  smaller  wheel,  the 
axis  E  of  one  of  them  is  placed  arbitrarily  with  reference  to  the  axis 
F  of  a  tooth  of  the  other,  whose  contour  is  derived  from  a  curve  El, 
determined  as  in  Fig.  260.  It  is  clear  that  under  some  conditions 
this  elementary  tooth  may  be  poiifted,  as  it  was  in  Fig.  261  ;  and  in 

Fig.  264  the  condi- 
tions are  so  selected 
that  this  is  the  case ; 
but,  as  before,  the  cy- 
lindrical pins  being 
given  to  the  large 
wheel,  this  pointed 
tooth  makes  its  ap- 
pearance on  the  other 
side  of  the  axis,  the 
direction  of  the  rota- 
tion being  unchanged. 
Hence  we  perceive  that  in  order  to  secure  receding  action,  the  larger 
wheel  must  drive,  no  matter  to  which  one  the  cylindrical  pins  are 
assigned. 

Had  the  wheels  been  made  of  the  same  size,  but  with  the  least  dis- 
tance between  the  axes  greater  than  the  diameter  of  the  cylindrical 


FIG. 


TEETH   OF   FACE   WHEELS.  289 

pin,  the  same  process  would  have  enabled  us  to  determine  the  form 
of  the  working  tooth  for  the  other  wheel  :  in  short,  the  construction 
is  general  for  the  relative  positions  of  the  axes  above  mentioned,  and 
Fig.  258  represents  only  a  special  case. 

441.  Miscellaneous  Arrangements  of  Face  Gearing. — It  is  not  essential 
that  either  the  pins  or  the  teeth,  because  they  are  turned  in  the  lathe, 
should  be  inserted  into  plane  surfaces  ;  and  we  append  a  few  exam- 
ples in  which  they  are  otherwise  arranged. 

Let  the  axes  in  Fig.  265  intersect  at  any  angle ;  let  the  velocity 
ratio  be  given,  and  suppose  the 
cylindrical  pins  *to  be  assigned  to 
the  wheel  with  the  vertical  axis. 

Draw  AB,  dividing  the  angle 
DAG  according  to  the  velocity 
ratio  as  in  bevel  gearing.  Through 
any  point  E  of  AB  draw  a  verti- 
cal line  as  the  axis  of  a  pin  ;  and 
also  a  horizontal  line  cutting  the 
inclined  axis  in  C ;  then  EC  by  FIG.  205. 

revolving  around  A  C  will  generate  the  cone  ECF.  The  teeth  of  the 
inclined  wheel  are  to  be  solids  of  revolution,  whose  axes  will  evidently 
be  elements  of  this  cone ;  and  they  may  be  fixed  in  the  surface  of 
another  cone  OPL,  normal  to  ECF. 

A  pin  of  the  vertical  wheel  is  shown  at  E  in  contact  with  such  a 
tooth,  whose  contour  may  be  determined  as  follows  :  First,  suppose 
the  cylindrical  pin  to  have  no  sensible  diameter  ;  then  if  the  vertical 
wheel  turn  through  a  given  angle,  the  other  will  also  turn  through 
an  angle  which  is  known,  because  the  circumferential  velocities  of 
the  circles  which  roll  in  contact  must  be  equal. 

Consequently,  the  relative  positions  of  the  axes  of  the  given  pin 
and  the  required  tooth  may  be  determined  for  any  phase  of  the 
action,  and  their  common  perpendicular  at  that  instant  may  be 
found. 

Having  repeated  this  process  as  many  times  as  may  be  deemed  nec- 
essary, these  common  perpendiculars  will  obviously  be  the  radii  of  the 
transverse  sections  of  the  required  tooth  for  a  pin  of  no  sensible  diam- 
eter. From  these  the  meridian  outline  can  be  constructed,  and  the 
contour  of  the  working  tooth  is  derived  from  this  in  the  usual  manner 
by  assigning  any  radius  at  pleasure  to  the  cylindrical  pin. 

442.  A  modification  of  this  arrangement  is  shown  in  Fig.  266,  the  pins 
being  fixed  radially  in  the  periphery  of  the  cylinder  with  the  horizontal 
axis  ;  but  the  process  of  constructing  the  tooth  is  the  same  as  before. 

19 


290 


MISCELLANEOUS  ARRANGEMENTS. 


FIG.  266. 


From  the  employment  of  the  conical  surfaces,  which  afford  the  most 
convenient  and  natural  means  of  supporting  the  teeth  of  one  wheel 

of  a  pair,  these  combinations 
have  been  erroneously  supposed 
to  contain  the  germ  from  which 
bevel  gearing  was  developed. 

But  a  moment's  study  of 
these  figures  will  show  that 
the  principle  of  rolling  cones, 
upon  which  bevel  gear  wholly 
depends,  is  not  here  involved 
in  any  way  whatever  ;  the  fun- 
damental idea  appears  to  have 
been  that  of  causing  the  teeth 
or  pins  to  present  themselves  to 

each  other,  at  the  instant  of  passing  the  plane  of  the  axes,  in  the  same 
relative  position  as  though  the  axes  were  parallel. 

But,  as  shown  in  Fig.  267,  the  same  methods  and  processes  are  ap- 
plicable also  to  the  construction  of  what  may  be  called  bevel  face- 
gearing,  the  axes  of  the  pins  coinciding  with  the  elements  of  one 
pitch  cone,  and  projecting  nor- 
mally from  the  surface  of  the 
other. 

Face  gearing    is    for  general 
purposes    practically    obsolete ; 
hence  it  has  not  been   deemed 
necessary  to  discuss  in  relation 
to  it  the  questions  of  limiting 
lengths   or    numbers  of    teeth, 
angles  of  action,  and  the  like. 
For  models  and  light  machinery 
it  is  sometimes  employed,  on  ac- 
count of  the  facility  of  forming  the  teeth  in  the  lathe  ;  a  special  con- 
struction would  in   any  event  be  almost  necessary  for  each  individual 
case,  and  this  can  readily  be  made  by  the  aid  of  the  preceding  expla- 
nations. 

443.  Screw  and  Face  Gearing  Combined. — An  example  has  already 
been  given,  in  Fig.  257,  of  the  use  of  rollers  for  reducing  the  friction 
which  attends  the  action  of  the  worm  and  wheel.  In  that  case  the 
screw  was  of  the  hour-glass  form  ;  but  the  same  expedient  has  been 
employed  in  connection  with,  the  common  variety,  as  for  instance  in 
Van  der  Mark's  hoisting  apparatus  ;  the  axes  of  the  rollers  projecting 


SCREW   FACE   GEAR. 


291 


radially  from  the  periphery  of  a  cylinder,  and  their  outlines  being 
the  same  as  those  of  the  spur-teeth,  conjugate  to  the  rack-teeth  cut 
from  the  screw  by  a  meridian  plane.  Since  the  teeth  of  the  wheel, 
whether  they  be  rollers,  capable  of  rotating,  or  not,  are  solids  of  rev- 
olution, this  arrangement  combines  the  peculiarities  of  both  screw 
and  face  gearing. 

In  Fig.  268  is  represented  another  combination  of  essentially  the 
same  nature.  Upon  the  upper 
face  of  the  disc  B  is  formed  a 
spirally  coiled  rib,  constituting 
what  may  be  properly  called  a 
plane  or  face  screw. 

The  section  of  this  rib  by  a 
plane  normal  to  the  axis  of  the 
wheel  C  should,  as  in  the  figure, 
have  the  form  of  a  rack-tooth, 
the  conjugate  to  which  is  the 
meridian  section  of  the  teeth  of  C. 
Thus,  the  distance  between  the 
corresponding  edges  of  the  adja- 
cent coils  being  equal  to  the 
pitch,  the  action  is  precisely 
equivalent  to  that  of  a  rack  and 
spur-wheel. 

444.  This  arrangement  has  been 
erroneously  represented  as  having 
the  rib  coiled  in  the  form  of  an  FIG.  268. 

Archimedean  spiral,  and  the  axis  of  B  lying  in  the  same  plane  with 
the  axes  of  the  teeth  of  (7. 

But  the  radiant  of  that  curve  does  not  cut  the  coils  normally  ;  and 
in  order  to  secure  perfect  kinematic  action,  the  spirals,  as  shown  in 
the  figure,  should  be  involutes  of  a  circle  whose  circumference  is 
equal  to  the  tooth -pitch,  and  the  axes  of  the  teeth  of  C  must  lie  in  a 
plane  parallel  to  the  axis  of  B  at  a  distance  from  it  equal  to  the 
radius  of  that  circle.  * 

This  is,  of  course,  upon  the  supposition  that,  as  in  the  case  selected 
for  illustration,  the  spiral  expands  at  the  rate  of  one  tooth-pitch  in 
each  convolution,  thus  making  the  action  equivalent  to  that  of  a  com- 
mon single-threaded  worm.  We  may,  however,  double  or  treble  the 
rate  of  expansion,  increasing  the  angular  velocity  of  the  face-wheel 
in  the  same  proportion,  and  introducing  a  corresponding  number  of 
intermediate  ribs  or  coils ;  if  this  be  done,  the  diameter  of  the  base 


292 


SPHERICAL   SCREW   AKD    WHEEL. 


circle  of  the  involutes  must  be  also  doubled  or  trebled,  as  the  case  may 
be,  and  the  position  of  the  plane  of  action  changed  accordingly. 

445.  Spherical  Screw  and  Wheel. — A  curious,  and  we  believe  a  novel, 
modification  of  the  preceding  arrangement  is  shown  in  Fig.  269,  the 
rib  being  coiled  upon  the  concave  surface  of  a  hollow  sphere,  forming 
what  may  be  called  a  spherical  screw  ;  the  construction  is  as  follows. 

The  spherical  involute  OFF  is  generated  by  a  point  0  in  the  plane 
ACB,  while  the  plane  rolls  around  the  cone  ACKto  which  it  is  tun- 
gent.  Since  the  element  of  contact  is  the  instantaneous  axis,  the 
tangent  to  the  curve  at  any  point  is  perpendicular  to  the  rolling 


FIG.  269. 


plane  ;  which  latter  cuts  from  the  sphere  a  great  circle  tangent  to  the 
base  of  the  cone.  Therefore,  the  points  G,  F,  E,  etc.,  of  the  curve, 
will  in  revolving  about  the  horizontal  axis  of  the  cone,  come  normally 
into  the  plane  ACB,  at  points  in  the  circumference  of  the  great  circle 
cut  by  it  from  the  sphere. 

Or  otherwise  :  let  the  cone  ^IC/Tand  a  disc  ACB  move  in  rolling 
contact  about  fixed  axes ;  then  while  the'  former  makes  one  revolu- 
tion to  the  right,  the  latter  will  turn  to  the  left  through  an  arc  PO 
equal  to  the  circumference  of  the  cone's  base  AK,  and  if  the  sphere 


SPHERICAL  SCREW   AtfD   WHEEL.  293 

revolve  with  the  cone,  the  point  P  will  trace  upon  its  surface  the 
same  curve  PFO  in  a  reverse  order. 

If  then  a  groove  of  this  form  be  cut  in  the  concave  surface  of  a  por- 
tion of  a  spherical  shell,  and  0,  P,  Q,  etc.,  represent  equidistant  pins 
projecting  from  the  edge  of  the  disc  A  CB,  it  is  clear  that  a  rotation 
of  the  shell  about  the  horizontal  axis  will  impart  to  the  disc  a  rota- 
tion about  an  axis  perpendicular  to  its  plane,  with  a  constant  velocity 
ratio. 

446.  Let  us  now  take  the  circumference  of  the  disc  as  the  pitch 
circle  of  a  wheel,  and  lay  out  upon  it  the  form  of  a  tooth  suitable  for 
driving  in  its  own  plane  a  pinion  in  inside  gear,  as  shown  in  the 
upper  diagram  in  the  figure.  It  will  then  be  apparent  from  what 
precedes  that  this  tooth  outline  will  be  the  normal  section  of  the 
thread  of  the  spherical  screw,  whose  surface  it  will  sweep  up  if  the 
disc  roll  around  the  cone  as  before  ;  and  that  this  screw  will  correctly 
drive  the  internal  pinion  if  its  teeth  be  made  solids  of  revolution  as 
shown  in  the  cut.  The  axis  of  the  screw  being  horizontal,  the  axis, 
XX,  of  the  pinion,  is  not  vertical,  but  perpendicular  to  the  plane 
ACS. 

It  is  hardly  necessary  to  point  out,  finally,  that  the  pinion  may  be 
placed  in  outside  gear,  and  the  screw  cut  upon  the  convex  surface  of 
the  sphere.  In  short,  the  construction  is  a  general  one,  the  face- 
screw  and  wheel  being  but  the  special  case  in  which  the  radius  of  the 
sphere  becomes  infinite  and  its  surface  degenerates  into  a  plane. 


APPENDIX. 


PROBLEMS  RELATING  TO  CIRCULAR  ARCS. 


1.  In  Graphic  Operations  there  is  frequent  occasion  to  rectify  circu- 
lar arcs,  and  to  lay  off  arcs  of  given  linear  values,  either  upon  circles 
of  given  radii  or  subtending  given  angles. 

In  such  cases  the  following  constructions,  which  we  borrow  from 
Prof.  Rankine,  will  be  found  extremely  useful  and  convenient  ;  the 
results  being  obtained  much  more  expeditiously  than  by  calculation, 
and  with  a  degree  of  accuracy  amply  sufficient  for  all  ordinary  pur- 
poses. 

L  To  lay  off  on  a  right  line  a  distance  approximately  equal  in  length 
to  a  given  circular  arc. 

Let  AB,  Fig.  1,  be  the  given  arc,  and  A  Hits  tangent  at  A.  Draw 
the  chord  BA,  and  produce  it;  bisect  AB  in  D,  and  set  off  AE  equal 


FIG.  1. 

to  AD.     About  E  as  a  centre,  with  EB  as  radius,  describe  an  arc  cut- 
ting AH'm  F\  then  AF  =  arc  AB,  very  nearly. 

II.  To  lay  off  on  a  given  circle  an  arc  approximately  equal  in  length 
to  a  given  right  line. 

In  Fig.  2,  let  AB,  the  given  line,  be  tangent  at  A  to  the  given 


29G  APPENDIX. 

circle.  Set  off  AD  =  \AB,  and  about  D  as  a  centre,  with  radius 
DB  =  |  AB,  describe  an  arc  cutting  the  given  circle  in  F.  Then 
arc  AF  —  AB,  very  nearly. 

III.  To  find  the  radius  of  a  circular  arc  which  shall  subtend  a  given 
angle,  and  he  approximately  equal  in  length  to  a  given  right  line. 

Let  AB,  Fig.  3,  be  the  given  right  line.  Draw  AG  perpendicular 
to  AB  ;  and  also  AH,  making  the  angle  BAR 
equal  to  half  the  given  angle.  Set  off  AD  = 
J  AB,  and  about  centre  D,  with  radius  DB  = 
}  AB,  describe  an  arc  cutting  AH  in  F. 
Bisect  AF  by  the  perpendicular  EC,  which 
will  cut  AG  in  C,  the  centre  of  the  required 
arc  AF. 

2.  Amount  of  the  Error  in  the  above  Proc- 
. — It  is  stated  by  Prof.  Eankinc  that  in 


D        NE  the   application  of   either  of  these  rules  the 

F:°-  3t  straight  line  is  a  little  less  than  the  arc. 

The  magnitude  of  the  error  is  given  as  about  7J¥  part  of  an  arc 

of  60°  ;  but  it  varies  as  the  fourth  power  of  the  subtended  angle,  and 

may  be  reduced  to  any  desired  extent  by  subdivision.     Thus,  for  an 

arc  of  30°,  the  error  will  be  TJT  x  -j1^  =  TIio ¥  ;  and  for  one  of  20°, 

only  Tffcj-  x  A-  =  Tsto«- 

Practically,  therefore,  if  the  given  or  required  arc  subtends  an  angle 
of  over  GO0,  subdivision  should  be  resorted  to  in  applying  either  proc- 
ess. 

The  first  two  rules  are  applicable  also  in  many  cases  to  other  curves 
than  circles,  provided  that  the  change  of  curvature  in  the  part  to  be 
dealt  with  be  small  and  gradual. 

GRAPHIC   CONSTRUCTIONS   RELATING   TO   TANGENT   LINES. 

3.  The  Determination  of  a  Tangent  to  any  curve  with  absolute  pre- 
cision, must  of  course  depend  upon  a  previous  investigation  of  its 
mathematical  properties. 

But  a  sufficient  degree  of  accuracy  for  most  practical  purposes  can 
be  attained  by  purely  graphic  methods,  very  simple,  and  quite  inde- 
pendent of  the  special  nature  of  the  curve. 

The  most  direct  and  expeditious  of  these  consists  in  finding  by 
trial  the  centre  and  radius  of  a  circular  arc  which  shall  sensibly 
coincide  with  the  given  curve  in  the  immediate  vicinity  of  the  point 
of  tangency.  This  is  more  particularly  eligible  when  the  change  of 
curvature  in  that  part  is  small  and  gradual,  as  in  Fig.  4. 


APPENDIX. 


297 


P 
FIG-  * 


If  it  be  desired  to  draw  the  tangent  at  a  given  point  P,  we  find  by 
trial  the  centre  C,  as  above,  and  the  required  line  will  be  perpendicu- 
lar to  CP. 

If  it  be  required  to  draw  a  tangent  in  a  given  direction,  or  through 
a  given  point  0  not  upon  the  curve,  the 
line  is  drawn  mechanically  with  the  aid 
of  a  ruler,  which  can  be  done  as  accu- 
ately  as  a  line  can  be  drawn  through  two 
given  points.  Then  in  order  to  locate 
the  point  of  tangency,  we  find  the  centre 
C  as  before,  and  draw  the  perpendicular 
CP,  cutting  the  tangent  in  the  required 
point.  » 

4.  This  tentative  method  is  not  very 
reliable  in  respect  to  the  exac.  location  of  the  centre  of  curvature, 
especially  if  the  curve  be  flat  ;  in  which  case  a  considerable  variation 
in  the  radius  may.  be  made  without  sensibly  affecting  the  curvature 
of  the  circular  arc.  But  the  direction  of  the  radius  of  curvature  is 
thus  ascertained  with  considerable  precision,  as  may  readily  be  veri- 
fied by  seeking  for  two  other  centres,  D  and  E>  of  circular  arcs  which 
shall  lie  respectively  within  and  without  the  given  curve,  and  deviate 
from  it  equally  on  opposite  sides  of  the  given  or  apparent  point  of 
tangency.  If  the  manipulation  be  made  with  care,  these  three  cen- 
tres will  be  found  to  lie  very  nearly,  as  they  should,  in  one  straight 
line.  And  this  fact  may  be  practically  utilized  in  dealing  with  a  very 
flat  curve  ;  in  which  case,  should  the  centre  C  fall  at  an  inconvenient 
distance,  the  centre  D  of  the  inner  circle  may  be  used  instead  with 
equal  confidence. 

5.  The  follow- 
ing constructions 
may  sometimes 
be  preferred,  al- 
though they  are 
less  direct  and 
more  laborious. 

"In  Fig.  5,  de- 
scribe with  any 
convenient  radius 
a  circular  arc  MN 
about  P,  the  point  upon  the  curve  XL  at  which  the  tangent  is  to  be 
drawn.  From  various  points,  as  A,  B,  etc.,  taken  at  pleasure  upon 
KL,  draw  through  P  right  lines  cutting  MN  in  a,  b,  etc.  From 


FIG.  s. 


298 


APPENDIX. 


\a 

\ 


K 


\ 


these  intersections  set  off,  on  these  lines,  distances  equal  to  the  corre- 
sponding chords  of  the  curve  measured  from  P,  and  in  the  same  di- 
rection ;  thus,  for  instance,  aa!  is  laid  off  to  the  right,  and  equal  to 
PA  ;  while  bb',  equal  to  PB,  is  laid  off  to  the  left,  and  so  on. 
Through  the  points  d,  V,  etc.,  thus  located,  draw  the  auxiliary 
curve  R OP,  cutting  MNin  0  ;  then  OP  is  the  required  tangent. 

Should  the  intersection  at  0  be  too  acute,  the  distances  aa,  IV, 
etc.,  may  be  made  twice  as  great  as  the  chords,  thus  changing  the 

direction  of  the  auxiliary  curve. 

6.  A  similar  principle  is  involved 
in     the     process     for     finding    the 

\  point  of  contact  when  the  tangent  is 

\          \  9       It          given,  as  in  Fig.  6.     Draw  any  num- 

ber of  chords  parallel  to  the  tangent, 
and,  through  their  opposite  extremi- 
o?       ~~  ties,    draw  in   opposite   directions  a 
series  of  parallel  lines,  which  may  or 
may  not  be  perpendicular  to  the  tan- 
gent.     Upon   each   ordinate  set  off 
from  the  tangent  a  distance  equal  to 
,          the  corresponding  chord  or  some  mul- 
^          tiple  thereof,  and  through  the  points 
Fro.  6.  thus  determined  draw  the  auxiliary 

curve,  which  will  cut  the  given  one  at  the  required  point  of  tangency. 

NORMALS. 

7.  For  the  General  Problem  of  drawing  a  normal  to  a  given  curve 
from  a  given  point  without,   no  solution,  by  graphic  or  any  other 
means,  has  yet  been  discovered. 

When  the  given  point  lies  upon  the  curve,  the  mathematical  prop- 
erties of  the  latter  may  or  may  not  be  such  as  to  enable  us  to  draw 
the  normal  by  a  direct  and  independent  construction.  But,  in  gen- 
eral, the  graphic  operation  in  this  case  depends  upon  the  previous 
determination  of  the  tangent,  by  one  or  other  of  the  methods  above 
described. 

THE   ELLIPSE. 

8.  First  Method. — About  the  centre  (7,  Fig.  7,  describe  two  circles 
whose  diameters  are  respectively  equal  to  the  major  and  minor  axes, 
^4  5  and  MN. 

Draw  any  radius  CE  at  pleasure,  cutting  the  inner  circle  in  D  and 
the  outer  one  in  E.  Draw  through  D  a  parallel  to  one  axis,  and 


APPENDIX. 


299 


through  E  a  parallel  to  the  other  :  the  intersection  0  of  these  lilies 
will  be  a  point  upon  the  ellipse. 

This  is  the  most  accurate  of  all  methods  of  constructing  this  curve 
by  points,  all  the  intersections,  D,  E,  0,  being  right  angles. 

To  draio  the  tangent  at  a  given  point. — By  reversing  the  above  proc- 
ess, we  find  that  the  given  point  P  would  have  been  determined  by 
the  radius  CGH.  At  G  and  H,  draw  tangents  to  the  two  circles, 
cutting  the  axes  produced,  in  L  and  R  respectively  :  then  LR  is  the 
required  tangent  at  P. 

Conversely,  to  find  the  point  of  contact :  if  the  tangent  be  given,  pro- 


Fio.  7. 

duce  it  to  cut  MN  produced  in  L,  and  AB  produced,  in  R.  From 
L  draw  a  tangent  to  the  inner  circle,  from  R  a  tangent  to  the  outer 
one,  and  from  C  draw  a  perpendicular  to  these  lines,  thus  determin- 
ing G  and  //;  draw  through  G  a  parallel  to  AB,  and  through  If  a 
parallel  to  MN ';  these  will  intersect  in  P,  the  point  sought. 

9.  Second  Method.— Let  C,  Fig.  8,  be  the  centre,  AB'the  major 
axis,  DE  the  minor  axis.  About  D  with  radius  A  C  describe  an  arc 
cutting  AB  in  the  foci  F,  F'.  About  F  as  a  centre,  describe  an  in- 
definite arc  with  any  radius  FH,  greater  than  AF  and  less  than  BF. 
About  F'  describe  another  arc,  with  a  radius  F'G  =  AB  —  FH  \ 


300 


APPENDIX. 


this  arc  will  cut  the  one  first  drawn  in  0,  #',  two  points  of  the  re- 
quired curve. 

This  method  is  not  eligible  for   the   ordinary  purposes   of  the 


FIG.  8. 

draughtsman,  since  it  involves  much  more  labor  than  the  preceding 
one,  and  the  intersections  of  the  arcs  are  in  many  cases  too  acute  to 
be  reliable.  But  it  is  of  interest  as  depending  upon  the  property  of 
the  curve  that  the  sum  of  the  focal  distances  is  the  same  for  every 
point  upon  it,  and  equal  to  the  major  axis  ;  thus, 

pp>  =  OF  +  OF'  =  AF  +  AF'  =  BF  +  BF'  =  AB. 


10,  Third  Method.  —  Upon  this  property  also  depends  the  operation 

of  drawing  the  "gardener's 
ellipse"  by  the  aid  of  a 
string  and  two  pins. 

Let  two  fine  pins  be  fixed 
in  the  drawing  board  at  F 
and  F'  ;  around  these  pass 
a  loop  of  waxed  sewing  silk, 
of  which  the  total  length  is 
AB  +  FF'  :  if  this  loop  be 
kept  constantly  taut  by  a 
pencil  P,  the  latter  in  mov- 


produce  FO  and  F'O,  and 


FIG.  9. 

ing  will  trace  the  curve. 

To  draw  a  tangent  at  any  point,  as  0 
bisect  the  exterior  angles  FOI,  F'OK,  by  the  line  TT.  To  draw 
the  normal  at  the  same  point,  bisect  the  angles  I  OK,  FOF',  by  the 
line  NN.  Obviously,  the  axes  cut  the  curve  normally  at  their  ex- 
tremities A,  B,  D,  E. 


APPENDIX. 


301 


11.  Fourth  Method. — Fig.  9  illustrates  the  principle  of  a  common 
elliptographic  trammel.  The  three  points.  P,  M,  N,  are  in  one  right 
line,  the  distance  PM  being  equal  to  CD,  and  PN  equal  to  CA.  The 
point  M  being  then  kept  always  upon  the  line  of  the  major  axis,  and 
the  point  N  upon  the  line  of  the  minor  axis,  the  point  P  will  at  all 
times  lie  upon  the  ellipse. 

This  method  is  extremely  convenient  when  no  great  precision  is 
required,  the  three  points  being  selected  upon  the  graduated  edge  of 
a  scale,  or  marked  upon  the  edge  of  a  smoothly  cut  strip  of  paper. 

To  draiv  the  normal  at  P  :  at  M  draw  a  perpendicular  to  AB,  at  N 
a  perpendicular  to  DE ;  these  perpendiculars  intersect  at  0,  and  OP 
is  the  normal  required.  For  MO  and  NO  are  the  planes  normal  to 
the  paths  of  the  two  moving  points,  and  0  is  consequently  the  in- 
stantaneous axis  of  the  whole  line  NP. 


FIG.  10. 

12.  Fifth  Method. — To  inscribe  an  ellipse  in  a  given  rectangle, 
Fig.  10.     Join  the  middle  points  of  the  opposite  sides  by  the  right 
lines  AB,  EF  \  these  will  'be  the  axes,  and  intersect  in  the  centre  (7. 
Divide  the  semi-minor  axis  CF,  and  the  half  side  GF  of  the  rectan- 
gle, into  the  same  number  of  equal  or  proportional  parts ;  through 
the  points  of  subdivision  on  CF,  draw  right  lines  from  B,  and  pro- 
duce them  to  intersect  the  lines  drawn  from  A  to  the  corresponding 
points  on  GF;  these  intersections  will  lie  upon  the  ellipse. 

Or,  divide  the  semi-major  axis  A  C  and  the  half  side  AH  in  like 
proportion,  and  proceed  in  a  similar  manner,  the  two  series  of  inter- 
secting lines  converging  in  the  extremities  E  and  Fof  the  minor  axis. 

13.  The  same  process  is  applicable,  when  it  is  required  to  inscribe 
the  ellipse  in  any  given  parallelogram,  as  shown  in  Fig.  11.     But  in 


302 


APPENDIX. 


this  case  AB,  EF,  will  not  be  the  axes.  They  are,  however,  conju- 
gate to  each  other,  for  each  is  parallel  to  the  tangents  at  the  extrem- 
ities of  the  other :  and  since  the  parallelogram  can  always  he  con- 
structed if  AB  and  EF  are  given,  we  have  thus  a  simple  and  ready 
method  of  constructing  the  ellipse  upon  any  pair  of  conjugate  diameters. 


PIG.  11. 

In  order  to  determine  the  directions  of  the  axes,  describe  about  the 
centre,  (7,  a  circle  upon  either  of  the  given  conjugate  diameters,  as 
EF:  its  circumference  cuts  the  ellipse  in  P,  and  the  supplementary 
chords,  PE9  PF,  are  parallel  to  the  axes. 

14.  Sixth  Method.— -To  construct  the  ellipse  by  means  of  ordinates 
of  the  circle.  In  Fig.  12,  let  it  be  required  to  draw  the  ellipse  of 
which  A  0  is  the  semi-major  and  A  E  the  semi-minor  axis.  Describe  a 
circle  with  radius  CR  =  AE,  and  divide  CP  and  OA  into  any  number 
of  proportional  parts.  At  each  point  of  .subdivision,  erect  a  perpen- 
dicular to  OA,  equal  to  the  ordinate  of  the  circle  at  the  corresponding 


FIG.  12. 


point  of  CP ;  the  curve  OE  thus  determined  is  the  required  ellipse. 

Or  CR  may  be  made  equal  to  the  semi-major  axis  BF ;  then  if  the 
radius  CP  and  the  semi-minor  axis  BN  be  similarly  subdivided,  the 


APPENDIX. 


303 


T 


FIG.  13. 


ordinates  of  the  circle  will  be  equal  to  the  corresponding  ordinates 
of  the  ellipse  NF. 

15.  In  a  given  ellipse,  to  find  the  conjugate  to  a  given  diameter. 
Let  PO,  Fig.  13,  be  the  given  diameter.    Draw  any  chord  EF  parallel 
to  PO,  and  bisect  it,  then  the  required  conjugate  diameter  JfJ^passes 
through  the  point  of  bisection,  and  TT,  the  tangent  at  P,  is  parallel 
to  MN. 

Otherwise :  draw  the  chord  EF 
parallel  to  PO,  and  also  the  diam- 
eter EG  :  then  MN  and  TT  are 
parallel  to  the  supplementary  chord 
Fff. 

16.  To  draw  the  tangents  to  an 
ellipse  from'  a  point  without.     Let 

A  A,  Fig.  14,  be  the  major  axis,  C  and  0  the  foci,  P  the  given  point. 
About  P  describe  an  arc  through  the  nearer  focus  C ;  intersect  this 
at  M  and  N  by  another  arc  whose  centre  is  0,  with  a  radius  ON  equal 
to  A  A.  Draw  OM  and  ON",  cutting  the  ellipse  in  G  and  H ;  then 
PG  and  P^Twill  be  the  required  tangents.  For, 

OH  +  HN  =  A  A  =  0//+  HC-,    .  • .  CJV  =  HIT ; 

also,  PO  =  PN,  therefore,  PH  is  perpendicular  to  ON,  and  bisects 
the  angle  CHN. 

Similarly,  PG  is  perpendicular  to  CM,  and  bisects  the  angle  CGM. 

17.  In  Fig.  14,  draw  PC,  PO;  then  the  angles  CPG,  OPH,  will 

be  equal. .  For  MN  is  a 
common  chord  of  the  cir- 
cular arcs  whose  centres 
are  0  and  P,  therefore, 
PO  is  perpendicular  to 
MN;  also,  PH\s  perpen- 
dicular to  CN;  whence 
the  angles  OPH,  CNM, 
are  equal. 

But  CNM,  in  the  cir- 
cumference, is  one  half  of 
CPM,  at  the  centre,  which 
again  is  equal  to  CPG  ; 
therefore,  OPH  =  CPG. 

18.  In  Chapter  XL 

i4-  (348),  the  argument  in 

connection  witli  Fig.  213  depends  upon  the  following  demonstration. 


304 


APPENDIX. 


FIG.  15. 


Let  the  foci  C,  E,  of  tlie  ellipse  whose  major  axis  is  RR,  Fig.  15, 
be  so  situated  with  respect  to  the  foci  D,  F,  of  the  equal  and  similar 
ellipse  whose  major  axis  is  BB,  that  the  right  lines  CD,  EF,  which 
intersect  at  P,  shall  be  equal  to  each  other ;  then  the  common  tan- 
gents to  the  two  ellipses,  HK  and  GL,  will  also  intersect  in  P. 

For,  first,  joining 
DE,  the  triangles 
DEF,  DEC,  have 
two  sides  of  one  re- 
spectively equal  to 
two  sides  of  the 
other,  and  the  third 
side  is  common, 
whence  the  angle  at 
F  is  equal  to  the 
angle  at  C\  and  in 
the  triangles  DPF, 
CPE,  the  angles  at 
P  are  equal,  whence 
the  angles  PDF,  PEC,  are  also  equal :  and  since  EC  —  DF,  we  have 
also  DP  =  PE,  and  FP  =  PC. 

Next,  draw  TT  bisecting  the  angles  DPE  and  FPC;  by  the  pre- 
ceding construction  (16)  draw  PG  tangent  to  one  ellipse,  and  find  the 
point  of  tangency  G  ;  draw  DG,  and  produce  it  to  cut  TT  in  A  ;  draw 
AE  cutting  the  other  ellipse  in  H,  and  join  HP. 

Then  the  triangles  DP  A,  EPA,  have  the  angles  at  P  equal,  the 
side  PD  =  side  PE,  and  the  side  PA  common  j  the  triangles  are 
therefore  similar,  and  we  have 

ALP  =  AEP  ; 
but   PDF  =  PEC, 
.-.     ADF=AEC: 

consequently,  since  the 
ellipses  are  similar,  HE 
=  GD,  and  the  tri- 
angles PHE,  PGD, 
having  the  two  sides  and 
the  included  angle  of 
one  equal  to  the  two  D 
sides  and  the  included 
angle  of  the  other,  are 
similar  and  equal;  whence  PH  is  equal  to  PG  and  tangent  to  the 


•ttp* 

\^V        OF   THE 


UNIVERSITY 


APPENDIX. 


ellipse  at  H,    and    also    the    angle    GPD   is   equal 
HPE. 

Now  draw  PK  tangent  to  the  ellipse  at  K\  then  (17)  the  angle 
FPK  is  equal  to  the  angle  GPD,  and,  therefore,  to  the  angle  HPE. 
But  FE  is  a  right  line  by  hypothesis,  therefore,  HPK\s  also  a  right 
line  and  tangent  to  both  ellipses.  In  like  manner  it  may  be  shown 
that  PL,  tangent  to  the  ellipse  on  the  right,  is  a  prolongation  of  PG. 

It  will  also  be  observed  that  if  xx  be  an  arc  of.  an  ellipse  whose 
foci  are  D,  F,  the  major  axis  being  equal  to  DP  +  PF,  and  yy  an 
arc  of  a  similar  and  equal  one  whose  foci  are  C,  E,  then  TT  will  be 
tangent  at  P  to  both  those  ellipses. 

THE   PARABOLA. 

19.  The  Parabola,  Fig.  1C,  is  a  curve  every  point  of  which  is  equally 


FIO.  ir. 

distant  from  a  given  point  F,  called  the  focus,  and  a  given  right  line 
DD9  called  the  directrix.  Hence  the  axis  OFC  is  perpendicular  to 
DD,  and  the  vertex  V  lies  at  the  middle  point  of  PC.  Draw  any 
parallel  to  DD,  as  RLP  ;  then  the  points  in  which  this  line  is  cut  by 
an  arc  described  about  F,  with  a  radius  equal  to  LC9  will  lie  upon, 
the  curve. 

To  draw  a  tangent  at  any  point,  as  P  :  Draw  FP,  and  also  PA 
perpendicular  to  DD  :  then  the  required  tangent  bisects  the  angle 
FPA. 

Otherwise:  let  R  be  the  given  point.  Draw  RL  perpendicular  to 
the  axis,  and  on  the  axis  make  VM  =  L  V ;  then  MR  is  the  required 
tangent. 

Second  Method. — The  parabola   may  be   traced   mechanically,   as 

shown  in  Fig.  17.     Let  DD  be  a  ruler  fixed  to  the  drawing-board. 

Let  a  fine  thread  whose  length  is  equal  to  LA  be  fixed  at  one  end  to 

the  point  Lon  the  vertical  side  of  the  right-angled  triangle,  and  at  the 

20 


306 


APPENDIX. 


other  end  to  the  drawing-board  at  F,  the  focus  of  the  required  curve. 
Then  by  sliding  the  triangle  along  the  ruler,  keeping  the  thread  taut 
by  a  pencil  P  which  always  touches  the  side  A  L  of  the  triangle,  the 
A  7*3  T,  motion  of  the  pencil  will 

be  so  controlled  as  to 
trace  a  parabola,  since 
PF  is  always  equal  to 
PA. 

20.    Third  Method.— 
In  Fig.  18,  let  V  be  the 
J-  vertex,  VO  the  axis,  and 
P  a  point  in  the  required 
curve. 

Draw  PO,  VA,  per- 
pendicular to  the  axis 
and  equal  to  each  other ; 
divide  VA  in  any  man- 
ner, and  through  the 
points  of  division  draw  parallels  to  VO  :  divide  AP  in  like  propor- 
tion, and  join  the  points  of  division  with  the  vertex  V.  The  inter- 
sections of  the  lines  thus  drawn  through  corresponding  points  upon 
VA  and  AP  will  lie  upon  the  parabola  VP. 
21.  Fourth  Method.— IK  Fig.  19,  let  Fbe  the  vertex,  VO  the  axis, 


FIG.  18. 


FIG.  19. 


and  P  a  point  through  which  the  curve  is  to  pass. 

Draw  PR  perpendicular  to  VO,  and  make  RO  =  PO.  On  the. 
axis  set  off  VL  —  VO  ;  draw  PL  and  RL,  divide  them  into  the 
same  number  of  equal  parts,  number  the  points  of  division  in  oppo- 
site directions,  and  join  the  points  correspondingly  numbered,  as 


APPENDIX. 


307 


1,  1,  2,  2,  etc. :  the  lines  thus  drawn  will  be  tangents  to  the  required 
curve. 

To  find  the  point  of  tangency  on  any  one  of  these  lines,  for  instance 
1,  1.  This  line  cuts  the  axis  at  N ;  set  off  on  the  axis,  VK  =  VN, 
and  draw  KB  perpendicular  to  V09  cutting  1,  1,  in  the  required 
point  B. 

THE   HYPERBOLA. 

22.  The  Hyperbola  is  a  plane  curve  generated  by  the  motion  of  a 
point  subject  to  the  condition  that  the  difference  of  its  distances  from 
two  fixed  points  called  the  foci  shall  always  be  equal  to  a  given  line, 
whose  length  must  be  less  than  the  distance  between  the  foci. 

In  Fig.  20,  set  off  OF  =  CFr,  and  let  F,  F',  be  the  foci :  set  off 
also  CA  =  OB,  and  let  AB  be  the  given  constant  difference.  It  is 
evident  that 

FB  -  BF'  =  AF'  -  AF=  AB ; 

therefore  A  and  B  satisfy  the  conditions  and  are  points  on  the  curve. 
With  any  radius  FO  greater  than  FB,  describe  an  arc  about  F  as  a 
centre  ;  then  about  F'  describe, 
with  a  radius  F'O  —  FO  — 
AB,  another  arc,  which  will 
cut  the  one  first  drawn  in  two 
points  0,  0',  of  the  required 
hyperbola.  Since  with  the 
same  radii,  arcs  may  be  de- 
scribed about  the  other  foci, 
it  follows  that  the  curve  is 
composed  of  two  equal  and  op- 
posite branches ;  and  since 
FO  may  be  of  any  length, 
these  branches  are  infinite. 
The  point  C  is  called  the  cen 
tre,  and  the  line  AB  the  Fl°- 20- 

major  axis,  to  which  the  minor  axis  is  perpendicular. 

To  dmiv  a  tangent  at  any  point  of  the  hyperbola,  as  P  :  draw  PF9 
PF',  and  bisect  the  angle  between  them. 

Otherwise  thus:  describe  a  circle  upon  AB  as  a  diameter,  let  fall 
PG  perpendicular  upon  the  major  axis,  and  from  its  foot  G  draw 
GK  tangent  to  that  circle.  Find  the  point  of  tangency  K9  and  draw 
KL  perpendicular  to  AB  and  cutting  it  in  L  ;  then  LP  is  the  re- 
quired tangent. 


308 


APPENDIX. 


FIG.  21. 


By  the  converse  operation  we  may  find  the  point  of  contact  if  the 
tangent  be  given. 

To  find  the  asymptotes.  From  either  focus,  as  F,  draw  tangents  to 
the  circle  described  upon  AB  :  find  the  points  of  tangency  R  and  8, 
then  CR  and  CS  are  the  asymptotes. 

23.  Second  Method.     In  Fig.   21,  let   C  be  the  centre,  AB  the 

major  axis,  and  0  a  point 
through  which  the  hyper- 
bola is  to  pass. 

Describe  upon  AB  the 
semicircle  ADB,  and  draw 
DCE  perpendicular  to  AB, 
also  DT  tangent  to  the  semi- 
circle. Draw  OS  perpen- 
dicular to  DE,  and  with 
centre  C  and  radius  CL  = 
OS,  describe  an  arc  cutting 
DT  in  L  ;  on  SO  set  off 
SG  =  DL,  then  CG  is  an 
asymptote  to  the  curve. 

Draw  any  line  JT7  parallel  to  AB,  and  produce  it ;  set  off  DM  on  the 

tangent,  equal  to  KI,  then  on  KI  produced  set  off  KN  =  CM,  and 

JVwill  be  a  point  on  the  curve. 

To  find  the  focus.     Produce  GC  to  cut  the  semi-circumference  in 

R,  at  which  point  draw  a  perpendicular  to  GR ;  which  will  cut  the 

major  axis  in  the  focus  F. 

To  draw  a  tangent  at  any  point,  P.     Draw  PV  parallel  to  the 

asymptote    HC,    cutting    the    other 

asymptote  GR  in  V.     On  GR  set  off 

VU  =  CV,  and  draw  PU,  which  will 

be  the  tangent  required. 

24,  Third    Method.       Given,    the 
asymptotes  and  one  point  in  the  curve. 
The  construction  depends   upon  the 
property  that  if  any  line  be  drawn 
cutting  both  asymptotes,  the  parts  in- 
tercepted between  each  of  those  lines 
and  the  curve  are  equal. 

In  Fig.  22,  let  CR,  CS,  be  the 
asymptotes,  P  the  given  point. 
Through  P  draw  any  line  at  pleasure,  Fio.  22. 

as  EF,  and  make  FG  =  EP  •  then  G  will  lie  upon  the  curve.     Any 


APPENDIX. 


309 


point  thus  found  may  then  be  treated  in  like  manner,  thus  :  drawing 
HGK,  make  XL  =  JIG,  then  L  is  a  point  upon  the  hyperbola  ;  and 
so  on. 

To  find  the  vertex.  Set  off  upon  the  asymptotes  any  equal  distances 
CM,  CN-,  draw  MN,  and  bisect  it  by  a  perpendicular  CO,  which  will 
be  in  the  direction  of  the  major  axis,  and  cut  the  curve  at  the  ver- 
tex F. 

25,  Fourth  Method.  Given,  in  Fig.  23,  the  major  axis  AB, 'and  P 
a  point  in  the  required  curve. 

Draw  PO,  BE,  perpendic- 
ular to  AB  and  equal  to  each 
other,  and  join  PE.  Divide 
PO  and  PE  into  the  same 
number  of  equal  parts,  num- 
bering the  points  of  division 
from  P  upon  each  line.  From 
A  draw  lines  to  the  points 
upon  PO,  and  from  B,  lines 
to  the  points  upon  PE ;  the 
intersections  of  the  lines  thus  drawn  to  corresponding  points  of  divi- 
sion, as  for  instance,  Al,  Bl,  will  lie  upon  the  hyperbola  required. 


THE   SPIRAL   OF   ARCHIMEDES. 

26.  A  Spiral  is  a  plane  curve  traced  by  a  marking  point  which  moves 
along  a  right  line,  while  at  the  same  time  the  right  line  revolves  about 
one  of  its  points  as  a  fixed  centre. 

This  fixed  centre  is  called  the  pole,  and  a  right  line  drawn  from  it 
to  any  point  of  the  curve  is  called  a  radiant,  or  radius  vector.  Sup- 
posing the  angular  velocity  of  the  revolution  to  be  uniform,  the  linear 
motion  of  the  marking  point  along  the  radius  vector  may  be  governed 
by  any  law  at  pleasure,  and  thus  an  infinite  variety  of  spirals  may  be 
produced. 

In  Fig.  24  both  motions  are  uniform,  and  the  resulting  curve  is  the 
well-known  spiral  of  Archimedes,  also  called  the  equable  spiral,  be- 
cause the  rate  of  expansion  is  constant,  so  that  the  distance  between 
any  two  consecutive  coils,  measured  on  a  radiant,  is  the  same. 

If  this  distance  or  rate  of  expansion  be  given :  with  it  as  radius, 
describe  a  circle  about  the  pole  as  centre,  and  divide  its  circum- 
ference into  any  number  of  equal  parts  by  radial  lines  ;  divide  the 
given  distance  into  the  same  number  of  equal  parts,  and  set  out  from 
the  pole  P,  upon  consecutive  radii,  as  1,  II,  III,  IV,  distances 


310 


APPENDIX. 


equal  to  one,  two,   three,  etc.,   of  these  subdivisions  :  the  spiral  is 
then  drawn  through  the  points  thus  determined. 

If  any  two  radiants  de  given : 
divide  the  included  angle  into 
any  number  of  equal  parts, 
and  the  difference  between  the 
given  radiants  into  the  same 
number ;  then  set  off  on  the 
first  dividing  radial  line,  a  dis- 
tance equal  to  the  least  radiant 
JI  plus  one  of  the  subdivisions  of 
the  difference,  and  so  on,  eacli 
successive  radiant  being  greater 
than  the  preceding,  by  one  of 
these  subdivisions. 

To  draw  a  tangent  at  any 
point.  The  tangent  to  this 
spiral  can  readily  be  drawn  by 
geometrical  construction  ;  for 
the  motion  of  the  tracing  point 
is  the  resultant  of  two  known 
components.  The  direction  of 
tho  circular  motion  is  that  of 
the  tangent  to  the  circle  through 
the  given  point,  of  which  the  pole  is  the  centre,  and  its  magnitude 
may  be  represented  by  the  circumference  of  that  circle.  While  mak- 
ing one  revolution,  the  tracing  point  also  travels  along  the  radiant 
through  a  given  distance,  which  is  the  other  component ;  and  the  re- 
sultant, representing  both  in  magnitude  and  direction  the  actual 
motion  of  the  point  at  the  instant,  is  the  required  tangent  to  the  spiral 
path.  It  is  not  necessary  to  set  off  the  entire  circumference  in  making 
this  construction ;  should  this  be  inconveniently  large,  we  may  use 
one-half,  one- third,  or  any  other  fraction,  reducing  the  radial  compo- 
nent in  the  same  proportion. 

27,  In  Fig.  24  PA  is  the  zero  line  from  which  the  circular  divisions 
are  reckoned  in  constructing  the  curve  in  the  first  manner  above  ex- 
plained. And  as  the  length  of  this  radiant  is  zero,  the  circum- 
ference of  the  circle  described  with  it  as  radius  is  also  zero,  but 
the  other,  or  radial  component  in  ihe  determination  of  the  tan- 
gent is  constant ;  the  zero  line,  therefore,  is  tangent  to  the  curve  at 
the  pole.  This  is  called  the  axis  of  the  spiral ;  and  it  will  be  noted 
that 


FIG.  24. 


APPENDIX. 


3U 


1.  The  length  of  the  radiant  varies  directly  as  the  angle  of  rotation 

from  this  fixed  axis. 

2.  The  lengths  of  successive  radiants  which  include  equal  angles,  hav- 

ing a  constant  difference,  form  a  series  in  arithmetical  progres- 
sion. 

3.  This  spiral  consists  of  two  infinite  branches,  curving  in  opposite 

directions,  and  symmetrically  placed  with  reference  to  a  line 
passing  through  the  pole,  perpendicular  to  the  axis. 

THE   RECIPROCAL   SPIRAL. 

28.  This  is  the  exact  converse  of  the  Archimedean  spiral,  the 
lengths  of  the  radiants  varying  inversely  as  the  angle  of  rotation  from 
a  fixed  axis. 

In  Fig.  25  describe  about  the  pole  B  a  circle  with  any  convenient 
radius,  and  beginning  at  the  axis  BA,  divide  it  into  any  number  of 


n 


FIG.  25. 


qual  parts  at  the  points  /,  77,  777,  etc.  On  the  radii  through  these 
points  set  off  in  their  order  distances  measuring  respectively  1,  •},  J, 
J,  etc.,  by  any  scale  of  equal  parts.  Draw  also  lines  from  B,  making 
with  BA  angles  equal  to  |,  J,  \,  etc.,  of  the  angle  ABI ';  and  set  off 


312 


APPENDIX. 


on  them  by  the  same  scale  distances  measuring  4,  3,  2,  etc. ;  the 
spiral  is  then  drawn  through  the  points  thus  determined. 

This  curve,  evidently,  makes  an  infinite  number  of  convolutions 
about  the  pole,  which  it  continually  approaches,  but  never  reaches. 

THE   LOGARITHMIC    SPIRAL. 

29.  This  curve  presents  to  the  Archimedean  spiral  the  contrast  that 
the  successive  radiants  which  include  equal  angles  form  a  series  in 
geometrical  progression  :  each  being  greater  or  less  than  the  preceding 
in  a  certain  constant  ratio,  instead  of  by  a  constant  distance. 

Hence  the  radiant  which  bisects  the  angle  between  two  others  is  a 
mean  proportional  between  them  ;  thus  in  Fig.  26,  if  the  angles  APK, 
KPH,  HPC,  be  equal,  we  have 

AP  iPKi:  PK-.PH. 

PK  iPH'.'.PHi  PC,  and  so  on. 

Had  the  radiants  AP,  PH,  and  their  included  angle,  then,  been 
given,  the  intermediate  point  K  would  have  been  found  by  bisecting 

the  given  angle,  and 
setting  off  PK,  a 
mean  proportional 
between  the  two 
given  radiants. 

If  the  given  radi- 
ants lie  in  the  same 
straight  line,  as^4P, 
PB,  the  construc- 
tion is  the  same — 
this  angle  of  180°  is 
bisected  by  CP  per- 
pendicular to  AB, 
and  describing  on 
AB  a  semicircle, 
cutting  this  perpendicular  in  C,  we  have  CP*  —  AP.  PB,  as  re- 
quired, therefore  (7  is  a  point  upon  the  spiral.  Toward  A  set  off 
PD  =  PC,  and  describe  a  semicircle  on  DB  as  a  diameter,  cutting 
CP  in  K  Bisect  the  angle  CPB,  and  on  the  bisector  set  off  PF  = 
PE,  then  F  is  also  a  point  on  the  curve. 

In  like  manner  the  spiral  may  be  extended  as  far  as  desired  :  thus, 
drawing  the  chord  CB,  bisect  it  by  a  perpendicular  cutting  CP  in  O, 
and  about  0  describe  a  semicircle  passing  through  C  and  B ;  this 


Fl°'?6- 


APPENDIX. 


313 


semicircle  will  cut  CP  produced  in  G,  which  will  lie  upon  the  spiral, 
since  by  this  construction  PB?  —  CP.  CG. 

30.  In  view  of  the  practical  application  of  curves  of  a  spiral  form 
in  the  construction  of  cams  for  communicating  definite  motion  to  one 
piece  by  the  rotation  of  another,  it  is  of  interest  to  note  that  by  set- 
ting up  the  successive  radiants  as  equidistant  ordinates,  any  spiral 
may  be  transformed  into  a  curve  capable  of  transmitting  motion  with 
corresponding  changes  in  velocity,  while  the  driver  moves  in  a  right 
line.     And  conversely,  any  curve  may  be  transformed  into  a  spiral 
possessing  analogous  mechanical  properties,  by  setting  out  its  equi- 
distant ordinates  as  the  radiants,  taking  care  that  the  successive  ones 
include  equal  angles. 

THE   HELIX. 

31.  The  Helix  is  a  curve  traced  upon  the  surface  of  a  cylinder  of 
revolution  by  a  point  which  moves  uniformly  around  the  axis,  and  at 
the  same  time  travels  uniformly  in  a  direction  parallel  to  the  axis ; 
the  rates  of  the  two  motions  being  entirely  independent  of  each  other. 

Thus  in  Fig.  27,  let  the  relative  motions  be  such  that  the  marking 


point  shall  traverse  the  distance  A1 M  while  going  once  around  the 
cylinder  ;  then  in  going  half  way  around,  it  will  travel  half  as  far ;  in 
going  one  quarter  around,  or  through  the  arc  AB,  it  will  travel  one 
quarter  as  far,  or  through  a  distance  equal  to  AL,  which  will,  evi- 
dently, bring  it  to  the  position  B'.  Intermediate  points  are  readily 
found  by  subdividing  the  arc  AB  and  the  distance  AL  into  the  same 
number  of  equal  parts  at  1,  2,  3,  etc.,  and  projecting  the  former 
points  of  subdivision  to  lines  drawn  through  the  latter,  perpendicular 
to  the  axis,  as  shown. 

If  the  cylinder  be  cut  along  the  line  A'M,  and  unrolled  into  a 
plane,  it  will  develope  into  a  rectangular  sheet  whose  length  is  equal 


314 


APPENDIX. 


to  the  circumference,  and  the  helix  will  develope  into  the  diagonal  of 
this  rectangle. 

To  draw  the  tangent  at  any  point,  as  B'.  The  curve  pierces  the 
base  of  the  cylinder  at  A1,  corresponding  to  A  in  the  end  view.  Pro- 
ject B'  to  B,  and  perpendicular  to  the  radius  BC,  set  off  BJ  equal  to 
the  arc  BA.  Project  J  to  T'  upon  NA'  produced,  then  B'T'  is  the 
required  tangent. 

By  projecting  J  to  J'  upon  a  perpendicular  to  the  axis  through  the 
given  point  B',  it  will  be  readily  perceived  that  the  same  result  would 
have  been  reached  by  compounding  the  two  motions  of  the  point,  the 
resultant  being  the  tangent. 

A  curve  analogous  to  the  helix  may  also  be  traced  upon  the  surface 
of  a  cone,  or  of  a  hyperboloid,  by  a  point  moving  uniformly  along  an 
element,  while  the  surface  at  the  same  time  rotates  uniformly  about 
its  axis.  (For  illustration  t)f  the  conical  helix,  see  Fig.  226  ;  the  hy- 
perboloidal  helix  is  represented  in  Fig.  243.) 

DRAWING    OF   ROLLED   CURVES. 

32.  In  Fig.  28  A  A  is  a  curved  ruler  fixed  to  the  drawing-board,  and 
BB  is  a  free  one  rolling  along  it.  Let  a  pencil  be  fixed  to  and  carried 


D 


FIG.  28. 


by  the  latter,  either  in  the  contact  edge,  as  at  D,  or  at  any  distance 
from  it,  as  at  E.  Since  P,  the  present  point  of  contact,  is  the  in- 
stantaneous axis,  the  motion  of  D  is  in  the  direction  DF,  perpendic- 
ular to  DP,  the  contact  radius.  DF  is,  therefore,  tangent  to  the  path 


APPENDIX. 


315 


of  D,  traced  as  the  ruler  BB  rolls  ;  but  it  is  also  tangent  to  the  cir- 
cular arc  whose  centre  is  D  and  radius  PD,  consequently  the  path  of 
D  is  also  tangent  to  that  arc. 

Let  the  arcs  PC,  Po,  of  BB,  be  equal  to  the  arcs  PC',  Po',  of  AA, 
then  cD  will  be  contact  radius  when  c  reaches  c',  and  oD  when  o 
reaches  o'.  If,  then,  we  describe  with  these  radii  circular  arcs  about 
c'  and  o',  the  curve  traced  by  D  will  be  tangent  to  those  arcs ;  and 
that  traced  by  E  will  be  tangent  to  arcs  about  the  same  centres  with 
aE  and  oE  as  radii. 

Curves  thus  described  by  points  carried  by  one  line  which  rolls 
upon  another  are  called  rolled  curves,  roulettes,  or  epitrochoids;  and 
the  drawing  of  a  series  of  tangent  arcs  as  above  explained  is  the  read- 
iest and  most  reliable  known  method  of  laying  them  out. 

The  line  which  carries  the  tracing  point  is  called  the  generatrix  or 
describing  line,  and  the  one  in  contact  with  which  it  rolls  is  called  the 
directrix  or  base  line  ;  either  of  these  may  be  straight,  or  both  may 
be  curved. 

THE   CYCLOID. 

33,  This  curve  is  traced  by  a  point  in  the  circumference  of  a  circle 
which  rolls  upon  its  tangent. 

In  Fig.  29  find  Aa',  the  length  of  a  convenient  fraction  Aa  of  the 
circumference  ;  step  this  oif  upon  the  tangent  the  required  number  of 


E 


FIG.  29. 


times,  making  A  E  equal  to  the  semicircumference.  Divide  each  into 
the  same  number  of  equal  parts,  draw  chords  from  P  to  the  points  of 
division  on  the  semicircle,  with  which  as  radii,  strike  arcs  about  the 
corresponding  points  on  AE\  the  cycloid  is  tangent  to  all  these  arcs. 


316 


APPENDIX. 


To  find  points  on  the  curve  :  "When  aC  becomes  contact  radius,  it 
has  the  position  a'R,  perpendicular  to  AE.  The  angle  aCP  remains 
unchanged ;  therefore,  make  a'RL  equal  to  it ;  then  RL  is  the  gener- 
ating radius  in  its  new  position,  and  L  is  a  point  on  the  cycloid. 
Also,  a'L,  the  instantaneous  radius,  is  normal,  and  a  perpendicular 
to  it  is  tangent  to  the  curve  at  L. 

Conversely :  Let  0  be  any  point  on  the  curve ;  about  this  as  a 
centre  describe  an  arc  with  radius  equal  to  CP.  This  arc  cuts  CD, 
the  path  of  the  centre  of  the  rolling  circle,  in  S ;  then  08  is  the  gen- 
erating radius  ;  Sbr,  perpendicular  to  AE,  is  the  contact  radius,  and 
V.O  is  normal  to  the  cycloid. 

THE   EPICYCLOID. 

34.  The  describing  circle,  in  Fig.  30,  rolls  on  the  outside  of  another 
whose  centre  is  G. 

Draw  the  CQmmon  tangent  at  A,  set  off  upon  it  the  length  of  A  a 


FIG.  30. 


(any  convenient  fraction  of  semicircumference  AP)  and  find  the  arc 
of  the  base  circle  equal  to  that  length.  Step  this  oif  as  above,  mak- 
ing AE  =  semicircumference  A  P.  The  curve  is  drawn  by  tangent 
arcs  in  the  same  manner  as  the  cycloid. 


APPENDIX. 


317 


The  path  of  the  centre  of  the  describing  circle  is  in  this  case  another 
circle  whose  centre  is  G ;  and  the  contact  radii  a'R,  b'S,  are  prolon- 
gations of  the  radii  Ga',  Gb',  of  the  base  circle  ;  which  slightly  mod- 
ifies the  processes  of  finding  the  point  of  the  curve  corresponding  to 
a  given  point  of  contact  and  the  converse. 

THE  HYPOCYCLOID. 

35.  Traced,  as  shown  in  Fig.  31,  by  a  point  in  the  circumference  of 
a  circle  rolling  inside  another. 

The  construction  is  in  all  respects  the  same  as  in  the  case  of  the 


FIG.  31. 

epicycloid,  and  the  diagrams  being  lettered  to  correspond  throughout, 
no  further  explanation  is  needed. 

In  all  three  of  these  curves,  if  the  rolling  continue  beyond  E,  anew 
branch  springs  up,  which  is  of  course  perfectly  symmetrical  with  EL. 

It  is  to  be  particularly  noted  that  these  branches  are  tangent  to  ED, 
and  therefore  to  each  other  at  E. 

THE   INVOLUTE   OF  THE   CIRCLE. 

36.  This  is  in  a  manner  the  converse  of  the  cycloid,  being  gener- 
ated by  the  rolling  of  a  tangent  right  line  upon  a  circle.  Thus  in 
Fig.  32,  the  ruler,  carrying  in  the  line  of  its  edge  the  pencil  P9  while 
rolling  around  the  cylinder  describes  the  curve  in  question. 

It  may  also  be  regarded  as  generated  by  unwinding  an  inextensible 
fine  thread  from  a  cylinder  :  the  thread  being  always  taut  arid  always 
tangent  to  the  circle,  its  length  is  equal  to  that  of  the  arc  from  which 
it  was  unwound  ;  thus,  beginning  at  0,  make  the  tangents  AP,  BE, 
DF,  respectively  equal  to  the  arcs  OA,  OAB,  OBD,  and  so  on  ;  then 


318 


APPENDIX. 


the  curve  passes  through  the  extremities  P,  E,  F9  G,  etc.,  of  these 
tangents. 

The  tangent  OG,  then,  will  be  equal  to  the  circumference  of  the 
circle,  and  if  the  unwinding  be  continued,  the  result  will  be  the  for- 
mation of  a  spiral,  the  distance  between,  the  successive  convolutions, 

measured  on  the 
tangent  to  the  base 
circle,  as  for  in- 
stance PH9  being 
constant  and  equal 
to  the  circumfer- 
ence. 

Considering  it  as 
traced  by  the  ruler 
as  in  the  figure,  it 
will  be  seen  that  as 
the  point  of  contact 
A  is  the  instantane- 
ous axis,  the  edge 
AP  is  normal  to 
the  curve.  This 
l>eing  true  for  all 
positions  of  the 
ruler,  we  have  the 
simple  construction 
Fl°-  32-  that  the  normal  at 

any  point  of  the  curve  is  tangent  to  the  base  circle. 

If  the  rolling  of  the  ruler  continue  in  the  direction  of  the  arrow,  it 
is  evident  that  after  P  reaches  0  a  new  branch  will  be  formed  as  shown 
by  the  dotted  line  ;  the  two  branches  being  tangent  to  each  other, 
and  to  the  radius  CO  at  its  extremity. 

THE   EPITROCHOID. 

37.  Although  the  term  epitrochoidal  is  used  in  a  general  sense,  in- 
cluding all  rolled  curves,  yet  custom  sanctions  also  a  special  sense, 
and  the  curve  traced  by  the  rolling  of  one  circle  upon  another,  when 
the  marking  point  is  not  situated  upon  the  circumference,  is  the  one 
ordinarily  meant  when  "  The  Epitrochoid  "  simply  is  mentioned  with 
no  qualifying  word  in  connection  with  it. 

In  Fig.  33,  if  the  circle  whose  centre  is  C,  roll  upon  the  circle  whose 
centre  is  D,  carrying  the  marking  point  P  situated  without  the  cir- 
cumference, it  describes  the  looped  curve  PLE,  called  the  curtate 


APPENDIX. 


319 


epitrochoid.  If  the  tracing  point  be  situated  at  F,  within  the  circum- 
ference, the  resulting  waved  curve  FJOTis  called  the  prolate  epitro- 
choid. 

Since  the  rolling  circle  measures  itself  off  upon  the  base  circle  as  in 
the  preceding  cases,  the  position  of  the  generating  radius  can  always 
be  found  as  in  the  construction  of  the  epicycloid,  and  its  length  being 
constant,  points  on  the  curve  are  readily  found  ;  and  the  instantane- 
ous radius  being  always  normal  to  the  epitrochoid,  the  tangent  at  any 


FIG.  33. 

point  may  be  drawn  with  the  same  facility.  For  example,  let  it  be 
required  to  draw  the  tangent  at  L  ;  with  radius  equal  to  CP  describe 
an  arc  cutting  the  path  of  the  centre  in  R,  draw  R G,  cutting  the  base 
circle  in  a' :  then  a'R  is  contact  radius,  RL  is  generating  radius,  a'L 
is  the  normal,  and  a  perpendicular  to  it  is  the  required  tangent. 

DOUBLE   GENERATION   OF  THE   EPICYCLOID. 

38.  By  way  of  distinction,  the  curve  traced  as  in  Fig.  30,  by  the 
rolling  of  one  circle  upon  another  in  external  contact,  is  called  an  ex- 
ternal epicycloid.  But  if  the  contact  be  internal,  the  curve  traced  by 
the  rolling  of  the  larger  upon  the  smaller  is  called  an  internal  epicy- 


320 


APPENDIX. 


cloid  ;  and  in  Fig.  34,  the  circle  whose  centre  is  B,  rolling  npon  the 
fixed  circle  whose  centre  is  D,  and  carrying  the  marking  point  F,  thus 
describes  this  curve,  of  which  FL  is  a  portion. 

Now  let  the  same  point  F  be  carried  by  the  circle  whose  centre  is 
C,  whose  diameter  FA  is  equal  to  the  difference  between  the  diame- 
ters FG  and  A  G  of  the  other  circles  :  it  will  then  trace  the  same 
path  FL. 

First,  let  the  three  centres,  C,  B,  D,  the  two  points  of  contact  A 
and  G,  and  the  tracing  point  F,  lie  in  one  straight  line  FG.  Then 
through  A  draw  EAH  in  any  direction  at  pleasure,  and  draw  .F/par- 


Fia.  34. 

allel  to  it.     Then  the  three  chords  EA,  AH,  FI,  subtending  equal 
angles  in  the  three  circles,  are  proportional  to  the  radii  ;  therefore, 

EA  +  AH  =  FS  +  SI. 

Also,  the  triangles  AHG,  FIG,  are  similar. 

Set  off  the  arc  GHK  =  arc  GL  Draw  the  chord  KM  =HA,  and 
prolong  it  to  L,  making  ML  =  AE.  Draw  the  diameter  MDT9  and 
a  parallel  to  it  through  L,  cutting  KT  produced  in  the  point  R. 
Then  the  triangles  MKT,  LKR,  are  respectively  similar  and  equal  to 
the  triangles  AHG,  FIG. 

Now  KD  bisects  MT  in  D,  and  when  produced  will  bisect  LR  in 
P  ;  therefore,  the  circle  round  LKR  will  be  tangent  at  If  to  the  circle 
round  J/LKT  and  AHG.  Consequently,  if  the  circle  whose  centre  is 


APPENDIX.  321 

B  rolls  upon  the  circle  whose  centre  is  D  until  J  reaches  K,  its  centre 
will  then  be  at  P,  and  KL  will  be  the  new  position  of  the  chord  IF ; 
the  point  ^meantime  tracing  the  internal  epicycloid  FL. 

39.  Produce  TM  to  0,  making  MN  =  NO  =  AC.  Then  the 
circle  whose  centre  is  .2V"  and  radius  NM9  will  pass  through  L,  and  the 
arcs  OL,  FE,  will  be  equal ;  because  M^L  =  AE,  and  the  angles  CAE9 
NML,  are  equal. 

The  point  Fwill,  therefore,  reach  L,  if  it  first  describe  the  arc  FE 
about  centre  C,  and  then  the  arc  EL  about  centre  D.  And  it  will 
be  perceived  that  this  is  equivalent  to  the  rolling  of  the  circle  whose 
centre  is  C,  upon  the  circle  whose  centre  is  D,  if  it  be  proved  that  the 
arcs  AM,  OL,  are  equal. 

In  order  to  do  this,  we  have,  first, 

BD  =  DP  —  AG  —  NL,  by  hypothesis, 
and 

LP  —  ND,  by  construction  ; 

.  • .  the  angles  ONL,  MDP,  are  equal. 
Draw  BI;  then  since  arc  KHG  =  arc  GI,  we  have,  second, 

KDG  iIBG  ::  BG  :DG; 

but 

IBG  =  2  (IFG)  =  2  (CAE)  =  2  (NML)  =  ONL  =  MDP, 
,  also 

KDG  =  ADP. 
Whence 

ADP  :  MDP  ::BG  :  DG  ; 

.-.  ADP  -  MDP  :MDP  :BG  -  DG  :DG, 
or 

ADM  :  MDP  ::  BD  :  DG, 


ADM  :  ONL  : :  ON  :  AD 

arc  AM  =  arc  OL  =  arc  FE. 


Q.  E.  D. 


40.  Although  the  epicycloids  thus  traced  by  the  rolling  of  the  two 
circles  upon  the  same  base  circle  are  identical,  it  is  not  to  be  assumed 
that  the  epitrochoids  generated  by  marking  points  not  in  the  circum- 
ferences of  the  describing  circles  will  be  the  same. 

On  the  contrary,  they  will  be  quite  different,  as  shown  in  the  dia- 
gram. If  the  generating  radius  BF  be  extended  to  W,  the  latter 
point  will  trace  the  internal  epitrochoid  WZ,  daring  the  generation 
21 


322 


APPENDIX. 


of  the  internal  epicycloid  FL,  the  final  position  of  the  generating 
radius  being  PLZ.  On  the  other  hand,  prolonging  the  radius  CF 
to  the  same  point  W,  we  perceive  that  during  the  generation 

of  the  external  epi- 
A  cycloid  FL  by  the 

r  o  1 1  i  n  g  of  t  h  e 
smaller  circle,  the 
external  epitrochoid 
WX  will  be  de- 
scribed, and  NLX 
will  bs  the  final  po- 
sition, of  the  gene- 
rating  radius  C  W. 
41.  Every  inter- 
nal epicycloid, 
tl\en,  is  also  an  ex- 
ternal one  ;  and  it 
may  be  remarked 
that  the  latter 
mode  of  generation 
is  usually  more 
convenient  in  prac- 
tical execution. 

Similarly,  every  hypocycloid  is  capable  of  two  generations.  Thus, 
if  in  Fig.  34  we  take  FIG  as  the  fixed  base  circle,  the  hypocycloid 
A  Y  will  be  traced  by  the  point  A,  whether  it  be  carried  by  the  semi- 
circumference  FSA  of  the  smaller  circle,  which  is  equal  to  FY,  or  by 
the  semicircumference  GHA  of  the  larger,  which  is  equal  to  GIY. 

PARALLEL   CURVES. 

42.  Parallel  Curves  are  those  whose  normal  distance  from  each  other 
is  everywhere  the  same.  If  one  curve  and  the  length  of  the  normal 
be  given,  the  other  is  readily  mapped  out  by  merely  describing  any 
number  of  circular  arcs  with  their  centres  upon  the  first  curve,  and  a 
radius  equal  to  the  normal  :  the  envelope  of  these  arcs  is  the  parallel 
curve. 

At  first  thought  it  is  natural  to  suppose  that  two  curves  thus  related 
will  be  similar  in  form,  like  two  concentric  circles.  And  this  will 
really  be  the  case,  if  the  derived  curve  be  exterior  to  the  first.  But  if 
it  lie  within,  that  is,  upon  the  concave  side  of  the  fundamental  curve, 
quite  curious  and  unexpected  results  may  arise,  of  which  Figs.  35  and 
36  are  sufficiently  remarkable  illustrations. 


Fro.  35. 


APPENDIX. 


323 


In  the  former,  the  original  curve  is  the  parabola  A  VB,  of  which 
VC  is  the  axis  and  0  the  focus. 

The  two  dotted  curves  resemble  it  in  form,  but  though  both  are 
symmetrical,  neither  is  a  true  parabola.  The  normal  distance  Vffis 
less  than  VO  ;  but  the  result  of  assuming  a  greater  one  is  the  forma- 
tion of  the  figure  of  which  the  construction  is  shown,  and  its  resem- 
blance to  the  original  curve  is  at  least  not  striking. 

In  Fig.  36^the  ellipse  of  which  the  axes  are  AB,  DE,  is  the  funda- 
mental curve  ;  and  as  before,  when  the  normal  distance  is  small,  the 
parallel  curves  are  somewhat  similar  to  it,  although  neither  is  a  true 
ellipse.  But  again,  upon  increasing  the  normal,  the  derived  curve 
loses  all  resemblance  to  the  original,  and  developes  the  four-cusped 


FIG.  36. 

figure  shown  within,  symmetrical  about  the  centre,  and  also  with  re- 
spect to  the  axes  of  the  ellipse. 

THE   LIMACON   AND   THE   PARALLEL  TO   THE   EPICYCLOID. 

43.  Mention  has  previously  been  made  of  these  curves,  whose  pecu- 
liarities, which  were  shown  to  have  direct  practical  bearing  upon  the 
true  theory  of  pin  gearing,  merit  for  that  reason  further  examination. 

In  Fig.  37  the  epicycloid  PBE  is  traced  by  the  rolling  of  the 
circle  from  C  to  D,  and  the  extremity  /  of  the  normal  PI  meantime 
traces  the  parallel  curve  IOHF,  consisting  of  two  branches. 

It  is  worthy  of  note  that  this  curve  will  always  exhibit  these  two 
branches,  however  small  the  normal  distance  chosen  may  be;  which 
will  be  readily  seen  if  we  suppose  it  to  be  traced  in  the  opposite  direc- 


324 


APPENDIX. 


tion  by  the  rolling  of  the  circle  from  D  to  C,  in  which  case  the  initial 
motion  of  the  circle  having  the  direction  of  the  arrow  at  E,  and  the 
point  E  being  the  centre  of  rotation,  it  is  apparent  that  the  initial 
motion  of  the  extremity  of  the  normal,  be  that  line  long  or  short,  will 
have  the  direction  of  the  arrow  at  F ;  so  that  under  no  circumstances 
will  there  be  an  interior  curve  similar  to  the  original,  as  was  the  case 
with  the  ellipse  and  the  parabola. 

Beginning  at  F,  then,  this  curve  descends  to  some  point  H,  and 
then  begins  to  rise,  the  normal  taking  successively  the  positions  EF, 
WU,  SH.  The  fact  that  there  will  be  a  cusp  at  the  lowest  point,  is 


FIG.  37. 

shown  by  the  consideration  that  were  there  either  a  wave  or  a  loop, 
there  would  also  be  a  tangent  in  some  direction  nearly  coincident  with 
US,  the  impossibility  of  which  is  perfectly  obvious. 

44.  Now  again  beginning  at  P  :  the  rolling  of  the  circle  is  a  com- 
pound motion,  consisting  of  a  rotation  around  the  travelling  centre  (7, 
and  a  revolution  around  the  fixed  centre  G.  We  may  then  suppose 
these  two  motions  to  take  place  separately,  in  succession. 

Thus  if  the  generating  circle  be  first  turned  through  the  angle 
PCR,  and  then  be  made  to  revolve  about  G  through  an  angle  meas- 
ured by  an  arc  A  0  of  the  base  circle  equal  to  PR,  the  tracing  point 


APPENDIX.  325 

P  will  have  reached  the  position  B  on  the  epicycloid,  and  OB,  the 
normal  to  that  curve,  will  be  the  new  position  of  the  chord  AR. 

"VVe  may  reverse  this  process  ;  if,  for  instance,  we  select  any  point  8 
on  the  epicycloid,  and  describe  a  circle  about  G,  cutting  the  genera- 
trix at  J,  then  AJ  is  the  chord  which,  by  the  preceding  operation, 
will  become  the  normal  ST,  and  the  arc  PJ  will  be  equal  to  the 
arc  AT. 

45.  A  series  of  chords  in  the  generating  circle,  drawn  through  the 
point  A,  then,  are  the  lines  which  will  eventually  become  normals  to 
the  epicycloid.     If  now  we  set  oif  upon  each  of  these  chords,  from  the 
circumference  toward  A,  a  distance  equal  to  P/,  as  er,  JK,  etc.,  the 
points  thus  located  will  determine  a  new  curve  IAKL,  called  the 
Limacon. 

And  just  as  points  upon  the  epicycloid  are  derived  as  above  from  the 
upper  extremities  of  these  chords,  so  points  upon  the  parallel  curve 
are  derived  from  their  lower  extremities.  Or  in  other  words,  as  points 
of  the  former  are  derived  from  points  upon  the  generating  circle,  so 
points' on  the  latter  are  derived  from  points  upon  the  limacon.  For 
instance,  the  point  g  of  the  epicycloid  would  be  determined  by  rotating 
P  about  C  to  e,  then  revolving  the  whole  generatrix  about  G,  the 
angle  being  measured  by  the  arc  AZ  equal  to  Pe  :  during  this  revolu- 
tion the  point  r  of  the  limacon  goes  through  the  same  angle,  giving 
the  position^  of  a  point  upon  the  parallel  curve,  which  lies  upon  gZ, 
the  final  position  of  eA. 

By  the  aid  of  the  limacon,  then,  we  may  determine  the  direction  of 
a  normal  at  any  point  of  the  parallel  curve  :  let  p,  for  instance,  be  the 
given  point;  we  first  describe  an  area  bout  G  through p,  cutting  the 
limacon  at  r,  then  draw  Ar  and  produce  it  to  cut  the  generating 
circle  at  e,  and  finally  draw  through  e  an  arc  cutting  the  epicycloid  in 
g}  then pg  is  the  required  normal. 

46.  To  Draw  a  Normal  to  a  Given  Epicycloid  from  a  Given  Point 
Without. — It  may  be  pointed  out  that  the  foregoing  indicates  a  method, 
circuitous,  it  is  true,  and  probably  more  curious  than  useful,  of  graph- 
ically solving  the  problem  just  enunciated. 

Supposing  the  epicycloid  given  as  in  the  figure,  and  any  pointy 
assigned,  the  normal  distance  of  this  point  from  the  curve,  even  if  not 
given,  may  practically  be  ascertained  with  great  accuracy  by  drawing 
a  circle  about  p  as  a  centre,  tangent  to  the  curve  ;  for  the  eye  is  capa- 
ble of  appreciating  the  fact  of  tangency  with  extreme  nicety,  if  the 
lines  be  fine,  although  wholly  unable  to  locate  the  point  of  contact. 
The  radius  of  this  tangent  circle  then  is  used  in  the  construction  of 
the  limacon,  by  means  of  which,  as  above,  the  normal  is  drawn.  It 


326  APPENDIX. 

may  be  added,  that  should  the  intersection  of  the  epicycloid  at  g,  by 
the  arc  through  e,  in  this  construction,  be  too  acute,  a  better  determina- 
tion can  be  made  as  follows  :  about py  wijth  radius  rA,  describe  an  arc 
cutting  the  base  circle  in  Z9  or  set  off  the  arc  AZ  equal  to  Pe ;  GZ 
produced  will  then  be  the  corresponding  contact  radius,  thus  locating 
the  position  of  the  centre  of  the  generatrix  ;  an  arc  about  this  centre 
with  radius  CP  will  then  cut  the  arc  eg  less  acutely  than  that  arc 
cuts  the  epicycloid  ;  and  if  gZ  be  found  to  pass,  as  it  should,  though 
the  given  point  p,  the  determination  may  be  accepted  as  at  least  accu- 
rate enough  for  all  practical  purposes,  if  any  such  there  be. 

47.  By  the  aid  of  the  limacon,  however,  we  can  determine  with  ab- 
solute precision  the  location  of  the  point  of  cuspidation  H,  and  of  the 
points  0  and  U  at  which  the  parallel  curve  crosses  the  base  circle,  as 
well  as  the  directions  of  the  normals  at  these  points. 

In  order  to  do  this  we  will  first  examine  the  limacon  itself  more  par- 
ticularly. If  we  suppose  CJ  to  be  a  crank  turning  about  C  as  a  fixed 
centre,  and  JK  to  be  a  rod  jointed  to  it  at  J,  and  capable  of  sliding 
endwise  through  a  socket  pivoted  so  as  to  rotate  freely  about  the  fixed 
centre  A  ;  then  if  JK  be  equal  to  PI,  it  is  clear  that  a  pencil  fixed  at 
the  end  A' of  this  rod  will  mechanically  trace  the  limacon  during  the 
rotation  of  the  crank. 

Now,  the  motions  of  the  points  J  and  A  being  always  known,  the 
instantaneous  axis  of  the  rod  can  be  found  for  any  given  position,  and 
we  are  thus  enabled  to  draw  the  tangent  and  normal  to  this  curve  at 
any  point  with  geometrical  accuracy.  The  motion  of  J  being  perpen- 
dicular to  CJ,  the  plane  normal  to  the  path  of  this  point  is  CJ  itself ; 
the  motion  of  the  point  A  of  the  rod  /JTat  the  instant  is  in  the  direc- 
tion JA,  therefore,  AQ  perpendicular  to  JK  is  the  plane  normal  to 
its  path.  But  since  both  A  and  /  lie  in  the  circumference  of  the 
circle  whose  centre  is  (7,  the  intersection  Q  of  these  normal  planes, 
that  is,  the  instantaneous  axis,  will  also  lie  in  that  circumference 
throughout  the  action,  and  will  at  any  given  instant  be  diametrically 
opposite  to  the  position  of  the  crank  at  that  instant. 

48.  To  draw  the  tangent  to  the  limacon  at  any  point.     In  illustra- 
tion of  the  above,  let  it  be  required  to  draw  the  tangent  to  the  given 
limacon  at  the  point  M.     Draw  through  A  the  chord  MA  and  produce 
it  to  cut  the  generating  circle  in  N,  and  draw  the  diameter  Nb  ;  then 
~b  is  the  instantaneous  axis  of  NMy  IM  is  normal,  and  a  perpendicular 
to  it  is  tangent  to  the  curve  at  M  as  required. 

49.  Now  when,  as  in  this  figure,  the  normal  distance  PI  is  less  than 
PA,  it  will  be  perceived  that  the  limacon  must  cross  the  base  circle 
of  the  epicycloid  at  A  ;  and  since  the  final  chord  AL  lies  outside  of 


APPENDIX.  327 

that  base  circle,  the  limacon  must  also  cross  its  circumference  again 
at  some  point  M  between  A  and  L.  And  further,  there  must  be  some 
point  K,  intermediate  between  A  and  M,  nearer  to  G  than  any  other 
point  of  the  curve  ;  at  this  point,  therefore,  the  limacon  must  be  tan- 
gent to  a  circle  whose  centre  is  G. 

Obviously,  the  point  0  of  the  parallel  curve  is  derived  from  the 
point  A  of  the  limacon  ;  and  making  AR  equal  to  IP,  and  the  arc 
A  0  equal  to  the  arc  PR,  we  may  find  the  direction  of  the  normal 
OB  either  by  describing  about  G  an  arc  through  R  cutting  the 
epicycloid  in  B,  or  by  producing  GO  to  cut  CD,  the  path  of  the 
centre,  in  X,  describing  the  generating  circle  in  that  position,  and 
making  the  chord  OB  =  AR  =  PI.  Also,  the  intersection  U  is  de- 
rived from  the  point  M.  We  here  observe  that  the  chords  MA,  AN, 
of  the  two* circles  which  are  tangent  at  A,  lie  in  one  right  line ;  there- 
fore, the  triangles  A  CN,  A  GM,  are  similar,  and  we  have  the  known 
magnitude  MN  divided  at  A  into  segments  directly  proportional  to 
the  given  radii  AC,  AG.  The  arc  PN  being  thus  determined,  we 
lay  off  the  arc  A  V  equal  to  it,  then  with  centre  F  and  radius  equal 
to  AM  describe  an  arc  cutting  the  base  circle  in  U,  at  which  point 
UVWis  the  normal. 

50.  Finally,  the  point  of  cuspidation  H  is  derived  from  the  lowest 
point  K  of  the  limacon.  And  this  point  must  be  so  situated  that  GK 
produced  shall  pass  through  the  instantaneous  axis  Q  of  the  rod  KAJ 
when  the  limacon  is  mechanically  traced  as  above  described. 

But  Q  is  then  diametrically  opposite  to  J,  and  P  is  diametrically 
opposite  to  A  ;  therefore,  PQ  is  parallel  to  A  J,  and  consequently  to 
AKi  which  is  a  prolongation  of  JA  ;  and,  moreover,  PQ  is  equal 
to  AJ. 

Then  from  similar  triangles,  A  KG,  PQG, 

AK        AG  AK       AG 

—    __  TCT  n  on  r*r*    —      —    "•— ™     

PQ  ~      PG'  AJ     ~  PG ' 

The  magnitude  of  A  J  being  thus  determined,  the  arc  AT  is  made 
equal  to  the  arc  PJ,  whence  the  position  of  the  contact  radius  TY 
and  the  normal  2JS  to  the  epicycloid  are  found  as  before,  and  8TH 
being  made  equal  to  JAK,  the  point  in  question  is  located  with 
geometrical  precision. 


FINIS. 


INDEX 


A. 

PAGE 

Action,  arc  and  angle  of 94 

"  line  of 43 

"  approaching  and  receding 96 

"  of  spur  and  bevel  wheels  compared 239 

Angular  velocity 4 

"  "  parallelogram  of 69 

Annular  wheels — epicycloidal  teeth 105 

"  "  involute  teeth 148 

"  "  in  pin  gearing 204 

Approximate  forms  for  teeth  of  spur  wheels 172 

«  «  «  « bevel  "  233 

Arbitrary  proportions  for  teeth  of  spur  wheels 103 

Axis  of  rotation,  instantaneous 14 


B. 

Backlash 107 

Band  motions 39 

Bevel  wheels,  pitch  surfaces 64 

"          "       in  double  pairs 66 

"insidegear 238 

"          "       teeth  of,  approximate  forms 233 

"          "          "      "  correct                " 236 

"          "         "      "  involute              "                                                              .  238 


C. 

Centres,  line  of 37 

Circular  pitch 94 

Classification  of  gearing 89 

Clearance.. 96 

Close-fitting  tangent  screws 267 

Composition  and  resolution  of  motions 12 

"           of  rotation  and  translation 19 

Computation  of  limiting  numbers  of  teeth 128 


330  IOTEX. 

PAGE 

Condition  of  compulsory  rotation 41 

"         "  constant  velocity  ratio 41 

Condition  of  rolling  contact 42 

Conical  lobed  wheels 74 

Conjugate  teeth 167 

Connected  points,  motion  of 13 

Constructive  mechanism 2 

Contact,  rolling,  sliding,  and  mixed ' 30,  35 

Contact  motions 40 

"            "       rate  of  sliding  in 42 

Continuous  motion 5 

Contraction  of  angles 51 

Cutter  engine,  pantagraphic 182 

Cutters,  series  of  equidistant . 186 

Cycle  of  motions 6 


D. 

Dead  points 38 

Degenerated  hyperboloids 85 

Describing  circle 93 

"            "     intermediate,  limiting  diameter  of 106 

"            "     interior  and  exterior,  limiting  diameter  of 107 

Determination  of  path  of  contact 170 

"             "a  series  of  cutters 186 

Diametral  pitch 176 

Direction 3 

Directional  relation .' 38,  42 

Dissimilar  lobed  wheels 54,  61 

Double  contact  of  teeth  in  inside  gear 107 

Driver  and  follower. .           8 


E. 

Elementary  combinations 7 

Ellipses,  rolling 51 

"        spherical 70 

Elliptical  bevel  wheels 72 

"        gearing,  construction  of  teeth 223 

involute  teeth 226 

pulleys 226 

Epicycloid  and  involute  compared 163 

spherical 233 

Epicycloidal  milling  engine 178 

Equidistant  gear  cutters 186,  194 


F. 
Face  and  flank. . 


INDEX.  331 

PAGE 

Face  gearing 287 

"         "       and  screw  gearing  combined 290 

Fallacy  of  Willis's  and  Rankine's  theory  of  skew  teeth 250 

Friction  gearing 46 

"            "     grooved 46 

"      bevel 65 

"            "     skew 87 

G. 

Gear  cutters,  equidistant  series  of 186, 194 

"        "         manufacture  of 178,  181 

Gearing  classified 88,  92 

Geometrical  method  of  investigation 11 

Graphic  representation  of  motion 10 

n. 

• 

Hindley's  screw,  or  hour-glass  worm 280 

Hollow  cones,  double  pair 68 

Hooke's  stepped  wheels 195 

"       spiral  gearing 197 

Hyperboloids  of  revolution 77 

"          in  internal  contact 81 

"          one  rolling  upon  another  which  is  fixed 250 

I. 

Inside  gearing 105,  148,  204 

Instantaneous  axis  of  rotation 14 

Interchangeable  spur  wheels ' 100 

lobed     "      58 

Intermittent  motion 5 

Intermediate  describing  circle 106 

Irregular  lobed  wheels 63 

Involute,  first  generation 144 

second       " 165 

spherical 239,  292 

teeth ._ 143 

' '     with  epicycloidal  extension 153 

"  •         "     original  pitch  circle 146 

"     proper  obliquity 146 

"     max.  pitch  for  given  obliquity 151 

*'     compared  with  epicycloidal 163 

"     for  elliptical  wheels 226 

"        Olivier's,  in  different  planes 253 


L. 
Least  amount  of  sliding  in  screw  gearing 278 


332  IKDEX. 

PAGE 

Limiting  numbers  of  teeth,  epicycloidal 114,  128,  134 

"      "     involute 153,162 

"  "        "      fl      in  pin  gearing 208,216 

"       diameter  of  pin      "    "        "       206 

Line  of  action 43 

"     "centres 37 

Links 8 

Link  motion,  velocity  ratio  in 36 

Lobed  wheels 49,  230 

"          "     derived  from  ellipse 52 

"          "     pin  gearing  for 231 

Logarithmic  spiral 47 

Low-numbered  pinions Ill,  149,  160 


M. 

Machine 1,  2,  3 

Mechanism,  pure  and  constructive 1,2 

"          train  of 8 

Mechanical  movements 6 

Milling  engine,  epicycloidal 178 

Mixed  contact 31,  35 

Momentary  constancy  of  velocity  ratio 38 

Motion  and  rest 3 

"     composition  and  resolution  of 12 

"     continuous,  intermittent,  reciprocating 5 

"     graphic  representation  of 10 

"      modes  of  transmitting 8 

"      modification  of 8 

"      of  connected  points 13 

"     of  a  rigid  body  in  space 25 

"     of  translation 18 

"      phases  and  cycles  of 6 

Motive  power 1 


N. 

Noise  and  vibration  of  incorrect  teeth 218 

Nomodont 139 

Non-circular  cones " 75 

"        wheels 49,  230 

Normal  component 13 


0. 

Oblique  rack  and  wheel 271 

"       screw  gearing 272 

"  "     and  rack .  376 


333 

PAGE 

Obliquity,  transverse,  in  skew  gearing • 82 

"          proper,  of  involute  teeth 146 

Odontograph,  Robinson's  templet 175 

Willis's 173 

Odontoscope,  Mac  Cord's 221 

Olivi£r's  involutes,  in  different  planes 253 

Original  pitch  circle 146 


P. 

Pantagraphic  cutter  engine 182 

Parallelogram  of  motions 12 

"             "  angular  velocities 69 

Parallelopipedon  of  motions 12 

Path 3 

< '    of  contact 151,  170 

"     "in  pin  gearing 217 

Phases  and  cycles  of  motion 6 

Pinions,  low-numbered Ill,  149,  160 

Pin  gearing 200 

"        "       for  lobed  wheels 231 

Pitch,  circular 94 

"     diametral 1 76 

Proper  obliquity  of  involute  teeth 146 


R. 

Rack  and  wheel ! 102,  147 

"      "         "     oblique 271 

"      "        "     in  pin  gearing 203 

"      "    screw 270 

Radial  planes,  wheels  with 209 

Rate  of  sliding  in  contact  motions 42 

Ratio,  velocity 9,  36 

Receding  and  approaching  action 96 

Reciprocating  motion 5 

Relation,  directional 38,  42 

Resolution  of  motions 12,  20' 

Rest,  absolute  and  relative 3 

Resultant 12 

"        motion  of  a  rigid  body 25 

Revolution  and  rotation 4 

Robinson's  odontograph 175 

Rolling  contact 32,  45 

"  "       condition  of 42 

"      cones 64 

"         "        in  double  pairs. . .  • 66,  68 

"         "        with  varying  velocity  ratio 69 

1  <      ellipses 51 


334  INDEX. 

PAGE 

Rolling  hyperboloids 78 

"  "  double  tangency  of 85 


S. 

« 

Sang's  theory  of  the  teeth  of  wheels 168 

"          "       practical  embodiment  of 269 

Screw  gearing 197,  266 

"          "      combined  with  face  gearing 291 

"          "       distinguished  from  twisted  gearing 91 

"      hour-glass 280 

"          "       least  amount  of  sliding  in 278 

"       multiple-threaded • 269 

"          "       oblique 270 

"          "      practical  proportions  o*f 268 

Skew  gearing,  pitch  surfaces 78 

"          "         teeth  of 242 

"          "        transverse  obliquity 82 

"          "         resemblance  of  to  screw  gearing 279 

Sliding,  rate  of  in  contact  motions 42 

Sliding  contact 80 

Spherical  epicycloid 233 

"       involute 239,  292 

"       screw  and  wheel *. 292 


T. 

Tables  of  equidistant  cutters 194 

"      4<  limiting  numbers  of  teeth,  epicycloidal 134,  135 

"      "        "  "          "     "        involute 162,163 

"      "        "  "          "     "        pin  gearing 216,217 

"      "        "  "          "     "        computation  of 128 

Tangential  component 13 

Tangent  screws,  close-fitting 267 

Teeth  of  bevel  wheels,  approximate 233 

"      "      "          "       correct 236 

"      "face        "     284 

"      "  lobed     " 223,228 

"      "     "         "       conical 240 

"      "  screw     " ^ 266 

"      "skew      " 242 

"      "      "         "      new  theory 253 

"      "  spur       "       conjugate 168 

"       approximate 172 

"      "     "          "      epicycloidal 92 

"      "     "          "       involute t 143 

"      "     "          "       Sang's  theory 168 

Three-leaved  pinion 113,  117 


INDEX.  335 

PAGE 

Transformation  of  rolling  curves 51 

Translation .'. , 18 

Transmitting  motion,  modes  of 8 

Transverse  obliquity  in  skew  gearing 82 

Twisted  spur  wheels  (Hooke's  spiral  gearing) 195 

"       bevel      " 241 

"       skew      " 265 

Two-leaved  pinion,  external 116 

"          "        internal..                                                                               .  122 


U. 

Uniform  periodic  motion 6 

Use  of  low-numbered  pinions 110,  149 

Unsymraetrical  teeth 171 


V. 

Velocity 3 

Velocity  ratio 9 

"         "    in  band  motions 39 

"         "     "  contact  motions 40 

"     ''link            "        36 

"          "    condition  of  a  constant 41 

"          "    momentarily  constant 38 

"         "    varying 47 

"          "    of  rolling  hyperboloids 79 


W. 

Wear,  effects  of  on  incorrect  teeth 219 

"      in  bearings,  effects  of 220 

Wearing  to  a  bearing 220 

Wheels  with  radial  planes 209 

Willis's  odontograph 173 


Table  of  Cutters  for  Teeth  of  Gear  Wheels, 


MADE  BY 


THE  PKATT   &   WHITNEY  COMPANY, 

HARTFORD,     CONN.,     U.     S.     A. 


All  Gears  of  the  same  pitch  cut  with  our  Cutters  are  perfectly  interchangeable. 


Diameter  of 
Cutters. 

Diametral 
Pitch. 

Price  of 

Cutters. 

Size  of  Hole  in 
Cutters. 

SET  OF  24  CUTTERS. 

For  each  pitch  coarser  than  10. 

5     inches. 

1* 

$25  00 

11  inches. 

No.    1  cuts                12  T 

41 

2 

20  00 

«        a 

No.    2     "                  13 

4 

2? 

18  00 

«        a 

No.    3     "  •               14 

31        " 

3 

15  00 

«     *  « 

No.    4    "                   15 

'Si        « 

Si 

12  00 

1 

No.    5     "                   16 

31 

4 

9  00 

«        «  i 

No.    6     "                   17 

3^        " 

5 

7  00 

"        " 

No.    7    "                  18 

3 

6 

6  00 

« 

No.  .8     "                   19 

21'       " 

7 

5  00 

«        « 

No.    9     "                   20 

21        " 

8 

4  50 

i         « 

No.  10     "               21  to  22 

2|        " 

9 

4  00 

«        «< 

No.  11     "               23  "24 

OL            « 

10 

3  50 

« 

No.  12    "               25  "  26 

2f 

12 

3  50 

«(                   H 

No.  13     »               27  "  29 

21        » 

14 

3  50 

tt 

No.  14     "               30  "  33 

2k        " 

16 

3  00 

"       " 

No.  15     "               34  "  37 

2 

18 

3  00 

" 

No.  16     "               38  "  42 

H       " 

20 

3  00 

" 

No.  17     "               43  "  49 

Ht 

22 

3  00 

«       (« 

No.  18     "               50  "  59 

11       •• 

24 

3  00 

« 

No.  19     "               60  "  75 

11       « 

26 

3  00 

u           « 

No.  20     "               76  "  99 

11       •< 

28 

3  00 

<«       « 

No.  21     "             100  "  149 

H       " 

30 

3  00 

«(       « 

No.  22    "             150  "  299 

11       " 

32 

3  00 

«          «( 

No.  23     "             300  Rack. 

No.  24    "                Rack. 

The  cutters  are  made  for  diametral  pitches.  By  diametral  pitch  is  meant  the 
number  of  teeth  per  inch  in  the  diameter  of  the  gear  at  pitch  line.  Two  pitches 
should  always  be  added  to  this  diameter  in  preparing  a  gear  for  cutting.  For  ex- 
ample :  a  gear  of  80  teeth,  8  to  the  inch,  diametral  pitch,  would  be  10  inches  on 
pitch  circle,  but  the  gear  should  be  turned  10$  (or  i).  The  teeth  should  always  be 
cut  two  pitches  deep  beside  clearance. 

The  cutters  are  made  for  a  clearance  of  -^  of  the  depth  of  the  tooth  ;  example  : 
8  to  the  inch  has  a  clearance  of  fiL4  ;  therefore  the  tooth  should  be  cut  two  pitches 
(i)  and  £t  deep.  The  gears  must  be  set  to  run  with  this  clearance  to  give  the 
oest  results. 

In  ordering  bevel  gear  cutters,  give  the  diameter  of  gear  at  outside  pitch  line, 
and  number  of  teeth,  also  the  width  of  face.  For  the  present  all  cutters  are  made 
to  order 


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